Physics-
General
Easy
Question
A smooth horizontal tube of length l rotates about a vertical axis as shown. A particle placed at the extreme end of the tube is projected towards 0 with a velocity
, while at the same time the tube rotates about the axis with constant angular speed
. The path of the particle is given by
![](data:image/png;base64,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)
The correct answer is: ![r equals l e x p invisible function application left parenthesis negative omega t right parenthesis](data:image/png;base64,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)
Related Questions to study
physics-
A disc of mass m and radius R moves in the x-y plane as shown in. The angular momentum of the disc about the origin O at the instant shown is
![](data:image/png;base64,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)
A disc of mass m and radius R moves in the x-y plane as shown in. The angular momentum of the disc about the origin O at the instant shown is
![](data:image/png;base64,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)
physics-General
physics-
Moment of inertia of a rectangular plate about an axis passing through P and perpendicular to the plate is I. Then moment of PQR about an axis perpendicular to the plane of the plate:
![](data:image/png;base64,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)
Moment of inertia of a rectangular plate about an axis passing through P and perpendicular to the plate is I. Then moment of PQR about an axis perpendicular to the plane of the plate:
![](data:image/png;base64,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)
physics-General
physics-
A square plate of edge a/2 is cut out from a uniform square plate of edge ' a ' as shown in figure. The mass of the remaining portion is M. The moment of inertia of the shaded portion about an axis passing through ' O ' (centre of the square of side (A) and perpendicular to plane of the plate is
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+TI1Uu9+sDEaoow3tzZhyl55gszJDys3I5DGKet5cWVrCmuvd4ezv4dCkVBVW2M+DCqNLAIv14e5Nv8LrSBhCJxGpCzGwzdQiMlH/Ww1wgj+W4PK8UMhCkHMViFJCEAR6Sh6a05p8c5MgJQ8vAS6QtrkikgB+QQjYNB+Qyi/iTCgaE5/SAzJP9ZppRgygjZNauP2TW/0+ZUp8SGOsFmjIwmpSUspZCF+wrgRS2K9nUSJQA40EoYtlt3CsgRxTqGUCWX77XBaFjn+WWgZHyz3qQFbm3f3lAG2ICMi2Y/KYjochSTkprs82m8UbYuVYQhk1Zcz9/F+I7r7ZjFIMSxB+jFNfh9SkqecchBEfUBUoeUUZTZqdUnV8YbtDlUsH2U5YmgoAzUSxwbQ+zgJJLNEpIH7T2NiGs5yUMIgNQ89w58lWAteWWOZCT2H58wEj5Zz0ov/DcXonSRfOvzz9nvCkPx0HpKGWzWYbnDYdAmUq5P8+tyYs85ivrcriOnXmkkQi4/Dr4ApGt3F43aV/gxhDQnk2a08SUUf7YBSVCF91LpZApym0mi6XtIuAoqHBcSVCe1wQll9d15Wc5msstSxxEjmgIIoAmWoQzUv5ZD4otFZT3K5M1W8KVz6zy42iarbHAIVAg8si7NkNwkWJ02ZnfmreXCvDK4ViiHOn5LBBilHj7vkw+mdeDElFGiaiGKaH8BpV1tsVxrRwSZEZbHT6H8FsYbmD9W/L2iB3e3uTKz7rSBlCrl9cL3btWAXrmZvsvgRt5/lkPi2nk9rbEtm4kT8coXWyiMW3atEH6mAP5GyE8Z1RSLvMoMpW360spddlRUilH6TqcQOT2HF9fSE9ipWqJqM4xJZRfaezf/u3fBnlQqU2Wtzd4PmeBS/l/gCfXXsnT8/glBRVRsMAxKU1ip4S8LITR71oXwAsIIyNCifcaLeyZtTC3cnuVI+mYf/PefmmkEFu3mxC7qTxSyMj9S7tGSRlyJr7WZec6DD7HYv1KRKi+H95e/4d1K33ge9Hdx2CVevmbmBLKz+NiWU1KDhPJIweZwVO3QdgUTK4llKVWXmF9hHIiiT5sqiYQZRMeu7bcMkBYXJfwlOrDI8Jv1kI/PMWW21MsdW6yhVCL5bDWVzTLsD6X+3MwvlNSrujWjNy8xAupJtUW3GgXRjjz9rXdpEvAE+lb8V7q/X34KCDnlvpOFiac8wudDEy8mHwrz5O6Xhh7H8t9Swt3WF4lEN5ey6doogusenRdEaZSVJKDcRJiemYCVwoFR8o/a6G7M3oyGHbeGZfDEJOPUohNuSKEV4JrghNRFhT1kYcaE+86uAVKmu8dwLDo9ozvl1LQEqJLlPyVDFYXRLWizz7p7MzAhJVf/mLgKBjPSUkpUhcBF628BkiIU+pdxg9gRfEFWNE+FjfW8DNIpXXaORgm6YhoAofQ1hBE6Rmo0cKeWQ9KRuGf9KQnDQjaZogtfYwQXlm51POBJIxuzXwdfQ4yG63ciDsyGVBSRPB6jtqK0xLcl5zUDFYXpD7LLLPMQKdwHJOBGVJ+zCzyhbVua4cM5OuxSyVBll++J5RrY0VzSB3c30TqLyilDjmCERbmK+V0PbMU4jnPeU7nsuLZEVNJ+W29Tmn1tiOTm2DQKWU00zQhfI/ysBWjNa8rDaSkjEOupGEQeHurSvsy+X3vW4N9LC1QEsU89alPnRphP+WX8xu0LoVDpsUKLSukoi0yR3j7IG66Qh/ET/x2AGNS2twhB2saTC7D0pVGOJ8hsl0Zi9y17mB2xFRQfh17yq9SL4Qah4JwDfhcCke59HyUlCu6NWNX5lKk6Dp4JU4Cz5Qvk+XtGQQNRdZ29M3R4zcn2u5bgxSTY3FP0aoIk1PKdy6e1Zgh5ffQbQRcEIA1Mg0Yj1wpm2WcEpwT23AhfUr9BTmE7JqLlFswuV2koRIf5hf7TFC85yjsnzXQy4GUU+UR1st5o8OPUlvhp1QrLG4ql4qO92MY2ro1896RfJMNFSC9HGEQXK8PGA67QYsS/NvVJdqEqIKhc1+kZlTDpswv9oTy1+qWFD0IQPl1yZqGUsrta51WOVyD8jImfbbhYlhio0TGomvBBSOSny8H1cMwqvPPOki7Qjaiw48i6/OnXJaDl5qyYvcdpcHS7jtg/mKTDUqaLycX/iNyw9v3BSMUZenS/v5diJ2uGLxmNDklSn1QU37/nzdclMp9zsn35rcMtwvCdKy/7/TpFAwmV1ogPeiClk8hp+ggZ5BHbP/kgbFm1J/4xCcO8vvSpi+8c3Rr+u2HfKFPjnzT17x3RMoZKznl2nn43waGRHSgv0X6ULtvDRxP154VnOWUCvtz5acgJo8C8eSl8Jon5e15b9colddy8ApSB8w8TqC2uUKA9+A1DDIisKu5AreQb6PcTDtGyj85ENJbOKPDjwcs/aBL7A8hEtNoVkK+nJyS5yF5VAJ4ex68L1QNoizd1TdSAqKbE9Ka3MZNSY05oCnB9gtdgMLl++6Xtl32QvH794SyT8cTtl8eRPG7OgUh+vCjBbMrt5d2GGwVCNFBKYUZKf/kgPJHKJ8TfqDkR6F5a/X70m5Q5tqcm/vmcnKEdNc+/CVEJcn3dPl1dYk2QcY5lOCmVAZKEI0I9xdaaKEB59HsUpxVmLDyy10oPwuc/6RSyZOHkokIWMEupWQoKDu+QB2+q+THumOEeXtC0rW8Mrq6TOq0adNaN1YcKf/kQdiP9NUWC2QpV+haxUVIre4fIbXrBMhn5PZ9GXlOICpJZHvYLbVEk7FprCi3LVpVQuawRBXLLrvswEFOFias/Ii3pz/96YMcSF20pHA8NYtnUljALsINDJTwBwko3O+aLJOLETa5SiddiJVhXVwAEoih0iJqYkbKP+uh4Quvo2eEt4/9IWo5MtnAz4jcLJ7JlZRDUH8nnziCrpWcgfhBFxGqdHDYbjuOyjvgvFQtao7O8wX3oK8Bh8Dw6VWZLExI+XXxaYlceOGFq0xpvoqqtPyyCQSewTLoBq9r8Y4QTH1Ybmdyu0I5TH6+sWLNECF2YutwS4cJ1Ej5JwcUXMcdT8igI2xr3j6WipMXDiZXUiXC+NUd3r4vYs8JvNWw880Q6UaVonJebXtWICARkVHCDEwJtp9iysW9QBMmQUjN28v/+9TtDVSQgH3q8Eo7Jld+WGKEm4itlro2Vgxih/AFscOIjcL+yQGnQm4WW2yxAflXav2mZNE7ggDOlUw653chokTIkPdBVJJ4a3zDsIQbbx8by0hXa9EC7iF2kVKCbD4f5RflTBZmKOeXhzXZft5eVCCk7lJi3liOE3X12uYKAaFSrPk3iCVGOIdJjF5wxE2tZVPKEr+/xvLmZZxZnfPfe/ed6Y6/3Z3ae8r+PoiMHniwfwcZzBrlfzDde9ft6fa77kutT/Pg/ene++5Pf++RY4P8nuLOP//8g3C9BFteBQFs0VdOAEv/hnEIYEy1eLsexWtb41ECJc+5qbZt5OLXfDSO1Z7PNXBXXXozszBh5ZdrBduPreTthWDTOgi0AGWSp8WuLCWmPYfOLM0VBlBY1wVKa1klsqhGRBrktt92h1mp/Pf87OS05Tr/kd52yLdSbf3h7dd/Px3zvu3TmzffJh3/w+Fqy7NC+W/+zsfTG9Z8cdr1xCtSecYeSrdeeXY6ZLct0hs22yeddW0/hpxH17orZM438ARKLgWjpE0CWC6v3i9iqDUElUDRXcs1rUIdNrdvrhGpfZ+D0mDEsTQbjZpwPaRfSQ5nBWZI+SmMBReY/PD2XWAooizIUOQrqkogEMoziBseoWsdgesjTQiD3L62+YJ0xP2dpy+g1iI8y5T/gVvScdusnOZfeLV0zEUlpb4jXXnG4WmT1Z6VHveYudJccz0pbfvFfptEBma68t/5s/TBjRZPcy/xhvSVawvm6oE/pO99bt/0quWfkh73OM+8Ujrson6bX4BIDellPgK6OO0WhQC2ejTvFJU7h7fvm9u7hzUbQnyrUPssFc9BTvLvtzWoRenSM7Z1A4pAGCC/g6GPoU+qPDMwYeXnMZdccsmBRevzO2SAvMk3OegCMjF+mcWqry4wRJHbl1Z+BfQolBjiEvoo/x03XJbOPOkT6eD99kq7vvvdaad37Zh22G6HtMveH0gfPfbUdOGv/vSoEPl353wwrfT4edOa+349NTPMB/9wefrMHq9Lyy+9cnr7nrulTddYfEyRFkrbndIvhw20Kf+9f7gqnXv6cemQA/ZJu++8c3r3TjuNPfP26d2775sOOeaE9O0rbkyP7KG8L/30xB3T0x+zcHrLpy9JTT9353XfS4dsvXZaZvm10w677Jpeu+J8A+U/4uL+BovhplCxk09s1qH2nStZTvYO4+3NITIvuKVhoWRt16CudnTkMwbf87V1IYIFbVIZcm4BmQhnsjBh5bf5Ae8t9++CsC3f+LCr5IcIsfDB4Gnq6GqucH25Oi8ud68ZouaCoD7ETrvyP5SuOeuwtNEqy6ZnPf/laYcPHJGOPe6zA8E6/tij0z7veE1a9knzpyVX3jKd9pM/jH9nDLf9PB34msXSvMtunr55Y57fPZSu++7xaeuXPy8tvfLG6Yizr04PpF+mj2y89JgiPSFtf2q/hSeBmvL//sITx+7xgvT0pV40ZlwOTp889vj0+RNOSJ897pPpoF02Tys9bd602PIbpI+d+8uHuYh7bzk/bbfC/GnhNfdJl9yep1D3pCvGIpSNV10mLb/21umkS8aU/Y+XpN1eOu9A+Y+8tP+PTJpH0RjyTQQQKWFe7uVBw9v32YEZNM1IJYJbGtbbC+m1o/s+Q9T2/Vipx2E1NyDNIeVU0ow2eCS6f2f7Uh9oYoicvw3NvfdLC3wC8nLn8NyRUnRBHR7xGF19pevjE5RxWGyWu6tFOEeX57/tVz9Ip37+s+mUb1yabv7rXWMh3L3pzr/+Kf3uxmvTRV8+Km26wjxjSvDU9K5TLx9TZHggXXnitmmxxy2c3nrsJeN/eyjdf8f16UuHbpNWWerZ6WVbfDhdcP34sucHf5z2XXfJgfJvd8rMCfvvvunK9LUvjCn8ly9IN/zxjnTv2DPfdcdf0i03XZ+uOPcLaae1nzJ2v7nTaz901jgXcVf6zofWT/POt3x631evnW4QHnow3XvrT9Lxe2+Sln/2c9JGu30mXfmH8c60G89N260+XfmPuPi3j4p6amC0EWI63eygk3eK5t5e7tx3BZ41G2Qp+jq6uKUmzHtswCnVra0pQRrnPxyatxU3kW9W4zcHoqNPb/+UWthTYyYRG/k67K6yi9w/79LKmdwSCIPlkK5fazICFjXKOCxtLbevoVX5H7wn3fKL89OxH9o9bf3WzdLrX/uatO4666T1NnhteuPbtk3v2W3H9JoVFxxTgmekd512+XSlufn8NO0F86aFX7FP+vE49/PQ7b9Ox2y9YnrsXHOlRVd5bdr9oCPSUR87Yiz3PTwd+aGd0yvHcui55povrfHWvdOBB30snX7B1WP+thtl5b8/3XbDpemLR+2X3rnF5mnTjTdM66+7Tlp3vQ3T69+8ddp1993TW9Z82tj9Fkwbf+TrySzcf80X0jpj0cBzNz86/Xrctj7wlwvTvq9mlB6Tllnrzel9h34sHfXRw9Ohh300Hbb3tPSSJece++yZaaMd35MO+Oin0lkXt0cAvLv5pPhC5lyuginn7c8555xHRAI1CLUpEm89zDbbAdxScFO1XYPAc/bpQgRpTR5BNKtbU6LOD6WFPQFsPIVBXpT23s9hIoV2LLMyYZ9WSi2RGm/kSbWuPqEaEkUZps9mojm8EyGznXRrnf/Wc9KWK84/JuRzpbV2/lz6xa2ZYbn71vTTs45OGy6D+HpG2vlLykAPpgsOfvWYB31uet/X/uHVHrjthnT64bumHXfePe01pqx7vWfPtPuYAuy2+x5pj13fltZY+klj15gnvXCjbdKOO+ybjjv7ytTHhBWV/94r0wEbPXvwzMu96ZD0g+uyfvMH/pp+deFpaZtVHz/2+fzp9YeeO5bp35lOmbZymvdpL0nHXPQPsvW+2y5Ln953j7TzruPPvOce0595LAffY9qmaaVnPG7sGoumtccMyg57HZhO/l638onIyAAFgVivj+xV2u1K/wLRMtvcracvGPzgjnBbNTAI+S5SbQ6L/ESLO90pYUorP28fTRYY+q46PMsXObhSYdeCBk0cognXF2LVoglhlYFUxlGG6eMpApF7MUaiEWxzdT3/X3+ejt1j4/SCxZ+Wnr/2ZmmXvfdP7//AAemggw5M+++7V9p5+83T2istMVC0tfY/M/3qks+nlz1xrrT8W45KN/auLP0k7b3WkwfX2OrUmVDqu/fG9NXDt0mrPeepaZnVNkrb77Fv2n/smQ886ID0/v32Sbu9a4u0wcuWH9zvOVt+PJ37pQPTCx4/T3rZrmek9p9SzXDLd9JWz2f0lkmHXd5/MQwOhkeUn1NgKYCjtl6/Cd4+FvD0WePRhKaifAPOtmgB10VG2roQQaQZW3hremtb0Dblwv4ATyk0U4vv02SR/3xXn11yYvMF0UStpVgIFmUYoX6fX/MJ4Aqi4UM+Fs8kEnHf6jPe/+f0i4vOS6efeGw65qiPpaOP+Xj61GeOTyefcW668vrfpd/+6qKxzz6fzvi/q9PV530+vf/AI9PZP330NmZVPPTndPm3vzQ23l9MP7p+uJSllvOnh+5M113x/fTVL342fWLsmY8ae+ZPfvrYdOJpX08X/eI36fc3/SSdeeIJ6ZTzfpjO++Kn0sEHH5Uu+F05vSvizpvTj87+wtgzfy39/E/9u+Uo37Rp0wbzR7H6NHIFpAbRMtsVbZbQN1rgEGKlHqUudSEGRBBR3cp3B65B38GUUX4LZJQ1goghaF099tGTbYL18ucrsErg7fPlmDVrLmRkSGIvgWGaNuK3ARAwSi15FcAESjGGIQlreOjvQyjQTEBV+YdBC0E7s0GRyAZyuK0unkO06R3Jh5bZvst1A/Lw+F1A0WStLwQZTeajRJz/Um8TZJoS8/bSgq5eFpGp9HXRRRedGu29ciE1WJaSZyz90k4OChULHyhaH68cK/ZEE352q8TkmzyrrzyLFKLPPgEBwsYA4QVqi4n0jlN+DUYzwwBMJmaK8k8ios6fN/nUQGFEmJEa9CUCc+CmpHRt8gVKxNIRDktK0lYiVqESEdMLlatSZ2kOTWwIa9HBk5/85EftZTArMWHll/M861nPGuT4XV13cvDYzw8JV6sQBDRtuC5rjPlFvJXAK8fKQWWYYYDIi408RAo1wdGl9ZSnPGVAOhGQqYSpqPz4FvxPGygMuSAffbo+m5BKaL4RrVoRWivLKevFrzkpEZPjGvLqVt8f+ogqAXIRpzRl6vxCoK46v8FTXuOVWfQ+v58fyx3l9jVlE1ZFC68csU93YQAjG9t8UY7SrkPACzAKJlPbZZ+9AmY3TFXlb9vHLm+iERkOC41BXQtsQGRKZqNE3OaworrF2/fZDJS8Kh+SLelBRBL/MnV+u/fKwaORoSv8Yb3D22vZrDVxCKvUU7taeEvIf7utrYRj4YiJl3uaJN8ZdsXX7IB/JeXH9ej6JB8Y+baW2RJiRagIzpjUoklGPy8Rt3n7XGbb+KgcGo2E+IjAZpVgytf5letima4fUOzTSqlvn7dvqxTIzzWBCNUIwTBNG4RK15VJ8t1apMCQYfxj4vWTm/yZtZmHsVL6qXWJBXgC5w2bwzYxK5TfGHm2Un9HDu/ovC6jn6Ok/JTRT7GFwjDgw4JMiSS1BbdxUyJTCmj+7Rng+WtwnWGqW+RVlCriVCUo7ec3pZU/9uqTg/u8S8hZ47CcmNpaWUcZRpgXYVUXZ5BDf7WQTKioBFT7riag4CXyiR9mMw9jQYC9B4ZXtGApsjXflNAzLLHEEp07x4pskD/qwthfZSJCj3jiXTS79DEOSlLDKr/xkVbhXQgshUCmGROsOCOtC6/2O3gBhtoW3JboWtdhvYO54Azk7OZeCtYk2TiMIL0YaSszORKpmrEdBrHAhnzhBuTlJXgO+wOYe2sKSsRvQMRgTDkhkURXKZIO5E1sbVWCKaX8hBRMCmtmkghsn736cstZW6BhoKPMZxL71nxB2qAEydJKI2phHi8bCyww/s1IBamo4hB8BaVzLQ1KJlLUIoVQcRD6WX6KILLFmd+d0ywT/y6++OIDYW5ji8EPRDJ0c889d3rMYx7z8DHvvPOmpZZaavA8xtkecCooSEnpEGNDsUJJeJqS8iuDUj58B4Mtd6bMOusItFVmiCilp8c9Tqfe9HfwDIsssshg3rtKarG5xgILLDD4nmv4d5555hmMg1ZdDDrijXHkPcmBphmOgNFkaDxL14+zlMDQ91lgIz2NlXrS0zaD6hnJK76gTymSjDDAZLBPE9uUUX7eSBjGK6t7Csv6rOfnUWIro7afOmIcCI/BHmaPdTBJhMsk5T/P1AShMvGU1cSXEBUFeSYjYoI8F0Pn12QJdd+D9Zd+UFqGqXRQBp6S0Cy4oHUB5Ws1D0oqqsBneF6Kz6C5X0DkwIjiM0Qzyq6UuXS92hH7Krhu6fnj4Pn9yIUfnyxdp3bYyecJT3jC4L9FP4z/MB5fnR7bTr6w77XvMoCRnjJCbevyGTqG3TVFqn34BpGOnhORcJ8mNjAvU0L5hTJYcAOiFbKPt9emGUpZ8/ZCWkYhwqq+q7dAZEDgfJcA1ry9UMyS3uAl2sI8ym8CPYvuMTuteG9euSm4XYf3NlZycfetHRpDhJ9Pe5oFNuVr1Q6hth8XVWHxAxh56Uhk4B386jBDIZIoXaPtYCgtlGGcSs8eB+MjOrHnQ+k6bQcDYG6UYRnnmoNoAgEcezBySjVI4WKlXte6ftUFUZhoq88OUry9RiXennHp8vYBKYlIhdGcLExI+b0gAoVlNuBduWeu0Lx+TaF5bArCs/Zdqx0QKnsm5FxbCSgmnmBJXbr4A8rPS4oSTJBcXmrgOpYrR7gsKpDfui5vKrzljSN0J9SMBqXoIsGE8CIp1/A914hrCaOF/sJZlQgeWElSXm77JwSlcJ4H9DyMYI54B57OO7mXyErqI0XgfQg7w8M4uK8jUhfv5pyaYQ14DiF8RC/5O8w333wD42OuGQrvQC6kGhTSWIpS5Px9vKx3ZZAoXNuPa1DESE8Zr7aGMHV6EYSUE9fRZYBwF6JH6YPUb5iGMFEug2VcZvsmn2BFTV4XeHvnyd9qeRKPLRw1ecKqvqu3gIDwpr4rNK81fOS8BAXsarkM5MrfBkYEcURIkHKUi2JRTKG88XrmM585UGqC3QaG0m8iSHlUJzy3pirP4rkZT2xxl1dhHFyrLxglfITxJ/yMHG9HqCkAj2b8NHchQtuAQKPgDIkIQDirEYvBZJyE0hQ7Jy4ZFIYlVvV1AcFqRyhGQwpqfGrwmXRIWtS2rt88qi6IIJR4zWEXlIZVKaRZ+IsuTidAVswP2ZWiuF9Xg9PMxISU32SZSMRPzXOaXKEyy0n4a0oprBKKUrBhmjbcN1hUHqZt4tVTnUPx+mwfloM3VdVoywlr0Bo8jCGb2ZA+4BemCkQgGHfkaRcYUEac4igN1spyDIzUhxwiWxGiNTBKznFNxqpLiRnfKA1blDPMsvHgpZDe4RQZPpHLZGHCOT/lL9X5AfvMawnfa95eGUlPs1SAV2lbHdWEHD2WAhOUmgfkHeVQJl5YWFu00Qar+uTIlIih0pJZO6QeuhIZGNGI0Nk7+p4DA61U6Tz/7V+ph9RJ+hDXyc/1385z5Oe0Hc5zMKhWKLpO6byuI67j3vEOrhXPHp/5t/ndtneI8/2/8RLVMFRSGilCl/czZn0W2IgyOCiRQVdDGEfCOXQ5koC0RjQ37LJxUVv8yrBUJ09rpgzbbwKayi9sixeTa9bqqsIqEyLPMUHDIF8K3FZ7R/jI/4VvhGyikJIg6KQuwkbC0TysbvQ8cj0GiQCrBHg/hFV+LoGlkPgJ56mZE1DXyM9zIBptONn8e9fhWr6H+MPL8C6l82bk8GzeWagq5C6dUztivDwjw2ocVAU4C3OGvygB14Qj4JnbyDSOBI/A6Mvb27y9vN/8mjdGp8vbS41i2bhQf5hl44yhKFcKXCIPp1x7b4C392JY0Vr4bhKCmPGSw4TE8io1X4Pe/MGGHLHhh4m3RLNPJ6Dn8DylzUiVJj2z1X8qGs1DNOEQ4UhB1LAJEaFw7zgPNyGPFrUoGyH/RAcii/x6Dt/DG+ghsMQYS988p3b4rvswjMJKpB3eoXTuRI54NikNT2W+ecrSuaWDDBgvwk/ZOQp5spDZtRnbZioZ6yxigU0b/xL7PmDORR018NTxY5p9y3G8ffRxWPDT1cQW8E7mnNwrXdZ2G6b8UyLsp/wsvgnTlEHZvFhJoQ1SvvNJn626Ann3FUvbllcxOqwqI9S2RDOH7xBgnj3/sQTCEaVJRq7Wpuw8IeOzn/3sAQvftmkDxZcb8vpSgzbhofhSJ8z3MC2ygIwjpFYjEroaLzNRiKp4fF66z5bqTZAFytnHcxqzINNUJGpGn9zpPyCHyOO2JiT8jdq+cih2vWtPCfdEQnoGpKcIpA/MGwMUzUZtUS6dEH1MCc9P4CkFJtqL1eqqiJk8rKoRMyU0m3BqyhzdfCYeyVizrDmkKCEszV9xJThh0JCVtXKTaEQYr54uKuHZShCGxh70xqK2khA8BwbYvYWuwzS4eCfVEvdRtjN2oqCJ4u/33ZPuGhP8u++5L90/ltJed8ON6QPv3z+9aJWV0n777jM0mcmA6qJkyEVSNcYdhiHT8oawtqYu1+SxY7emPou1cArSk+gJ6ONQQKQXP/iqClWbR5Eq3gqHwVjHbxZMBmaI8JPTCjFLlpNHFKqpexq8tvy8CZY2ynKIvTZLqx8gGocwqH0QwiJCaApLtHC2lSZ5bJNksgin96yBF46+AmPWBv377ut9anlvDd7Jdz27FAyw4UP19t/5u3TZeV9Jxx/94bTXTtPSFm9/W5r2zncOPOk7d9glLb3cCmmuuRdKHzx6uB+84AGtlKN0eg+6lE54LQR2vrSnpnCMXfSPMNZttXgRBiMS5biuaKopgzXDXgIyE6fFObaVCkWaImHlRySosZkSO/lEzl+amLDwPJBwaZgttWJApAhtymLiCSXLyuv38fY8OM9IWHh7uVjA9/uQlYwY4g9JRblqXtw7h/B0NZQwJvGDoniKYboaPadoJxQgfydRRi/lf+i29MNTD0tvX3/1tOTiS6YXvnS9tPXO+6T/OuaTafc935MWmH/etPATF0hbb7pues1LV0rLrbhW2mTHj6Szruge81zpKHJbCpKTacpebSkBAxelsraGMNek7BFBMCxdwGEgJclgiQeqAZ+hamEelRdrFSyGJeZbaTEMy5Qj/PLQjSGw+ou3F1ZZNNEXogdkB1IEwdZmadWDeW25et+cE8FEUAhMU1h4//D2tRZOkYwJjdxeLudvJQgVGQjCo6GkLVTkBRFOykyqIMPAc4t4vFdph9teyn/f79KZH94sPfsJc6XHLLleOuT0C9ONf5nORXx9TMFe9cpXpreP5cc/vuzS9ODdf0k3XfWj9MmdX50WHDN+izz/dekT55WZ9FzppERdG7mQFTJDdtrItDDSYfRzY9cE4+GaIghy2eWEMP16BsggpWxLz5pgJKJU2Labr+hOxBzlx1yGplypLxBsvMkepssJMLQGJMpybY1DPKOJV3fvk3PyjPJ2C1J4SBFDwH8j1HhN57R5e4ZG2YxQ1ASOAePtCY+oQN5XA+G2Is94IZH6dhyC544oxXPXwt1u5X8o/eb8I9Jq8+qpf3764DemP8Nvr782bTkW8gch9igP9uDN6TNvX3EQ/TxznfemH/7pkUaQQRPCejfv2MbzkBP3cK82Mo0BjYoSI93WEOaaOAURRGmlZgkiTiVXCkwp+5Kk5thcm3OcVi2393ccjjlTfiz1nEwp5efdCDyryrp6+C4Ln4MSqRAYkLZwl3VkFII1bVu0kSNye96x6RkjbPR5TZAYFxOqP134roJQ8/bY2vD2vECbt6fo8kgCj9Tqyj9zRNTTpQDQrfx3pguO3HjwS0GPWWmH9NWf3p5OPuHYh0PktrD75lN2SE8e+95cT39FOu5H0xl4siBs914WJnV1RSqvqfcHmVYbB+G0qgWjT17a9uzLr6knpGtszXE0gtWUsgTGIRqDupqNODeVEa3O5LgmQ1NG+XVtefEYaBa+TeCboIzCVYJc26sPKEps32SSSuRiE0JDQu87CKHcoyMoeU2RAEEq7agCyjLqxbrORAd5xJCDV4xQUS99W0MJ8CohMMOkRUqqfojCc4t6+lQBupX/3sHvBi7+2LnSY5d6VXrxBpunl7/0RYO5rAlo4Mqj35AWGFP++Z73xnT2TWPe/vLLBrKgMtNGgIKcF8NtzLrINMpCaRh9SlSDa6qouKYIos8uzsi4aARD0vUFo5Z3mNZSFBFZbFZj7rq4HNGSd5gsTEj5DbS8XAOJGqyQvy8oEWKKEFPMmlKBTTIwoX0XWEAsJHI0c3tGposriJZg5TtC11Y6EirqdaDMbXsCAkFAUHpv3MEwJU+CSUAJapsCBBg4xovyayFtwy1XfyutsZif0x7z4ouskM68uPu34W+97KS06XJPSHPN94y0wb7/m/Y/+lNppRX6bduWk2lt+z/wwLEuv61UBhF1cUK8cRcYbAaUUirH9fX2EOv0uypYqjVRUYrqSxekKAyo7cGHIcknigkpP6EXGgr1a1avBCGrBhfKJySvgZeL3U1t39QW5gWEb1Ef5+1yK5vnyG0dVnZ8oWSWnjqv1igip8TW8jLY3S5iiKFhjITqw7QzE0pG1nO3LVXNEemRscBPIMVq8Czrb7Bh2uBVq6ZtN1wpLbbIEmmV9bZO+x91cjr/x79Of/rbPQNC9/77702333JduvjcU9Ohe26V1vq3p6YFF105rf7m/dIqr3xdetkaq46lgSe1htg5mWZu2zwz4pPRl4O3Gf38mvLurtWSIFUSRZDDtg7AJnAR0RikguXeJZgjBpccihgZ4j7gdHBolk77/sxuzCphQsrPCiNpTE4faAeNhgdNJzWCiiFhuWPRRteyURCe8vCsrDQir81LQ3h7E23Ca9s5GXgTZdsqQte27RPPlS8WaZskHkYUQXkpY1sdOgeF0wHIO2KPGdsuMBQxxgwFPsVS2lLYr+QZDU7v3nmXsfm5Md17283p0rOPT3tu9Z9p7Zesnv59lVXT6mu8JK01NqZrrfnStPpqq6YXrbZGWmejN6W37nZYet1WO6flVlglvXvH7dJ117bXwINMM7ZtY8box6o6xrXN6OcEnairrWEIXCtCcBFY33IqD5yv06+laiGHEXUO06fhXUQu1nmIiifD68OEc36kVrPU1wRPEKG7iWrbeZXVDhJM3lNbtJEj78RjLXPPyOAESdSWI/MEcje99pjz2sBTGGQMwcRBuH4bEE9SAkbCgo6+EEXoCzAOKgdd6YExplDuY4xzQ1HK+TUxMZTIziKB+eD96c47/pJ+f8M16bILv5fOG4vQvv2d89Plv7h2LBK4O337uxekDTbaOK251prp9NNOafX2uBYhuzETwtfGjDGQAjB2lKDN6LtmEHTmoU/IThFFm10dgE3oKtRvEI1BNdnwXnnVqE+EBs7zLuSP/HJQffcymBmYsPKX6vw5KDPP48XkuLUwiXfOF21YMNIH8iiTSZCbtXnKFpWBmhc3YTyAXXa6wnFGy7tKCShaG4yHHnTCKbztm09SQuMgn1T379MRyVBEU4n0qDnGlD/CfkSk9yWgTUPZB/fcc/eg8iHd22H77TqJzZzh1r1WQ5BnjB3yrM3rxUrNrmsGRD/eNcq8tVJuEyJQDUneVeNNWwUrUlkyNMw6hyAbvQvyHJTKp2yTT8BqqQjd29ZGIwqRHCyrNKIPf2BCcyub990L5bCqlIG3rzH54SmFWW2GSWUhXxrapcgE2QTiQroY7xwWryiNUQBLVfuEfX3SAsaX8IcxJGwTWYgz4AbWX//hVKcNEV6bH+XbtggJeRar6tpKZeZRmtLnmoHYLGPYEFwHINY9GoNqFSy5vDTLM9mPorb+owmpoPN9r/kz4lOuySdXfkyvMgtlJsSqAiUYUPXt6A3o03IJ0eRRYutZz64+AMZBTqluTxHalnES+Gg8CsvcBqv5CDLB6dvrYByG7YjUVCItEEpLC9pYcMKlG5HSMnLDenvGL7ou237sJIBv4QFFZG3lW0aSnISxqxlfiDUL5r1P+oTMlQpSrq4OwBwRsUVjUNscRu8++ehTfQk4l3yS0xLZOKWUnzcAnipv9GlTZnlUeMfmz2HXQGmDrefRc+KMBQ5vz0uXBFxOyVPaOMLutvKqWlRAmWI/ANftYms9m3X50ps2g9cE4co7Iru+RzhPOumkQUSlzNTmKfEljBEeg0FqazetgfFjIMP4deX2IgxjJryuKRw50brbx9iZR8rrmpS5i6BjSDmHqCbV2rRL8ByxelTEVuObovrimfpWX4CsRWkRD1WrIk0Z5UfQILQIIVKEMlu5VQvdCbd1+Syr8/u0XAIPwlJquMmtrBw5GkBY4FpJSAMJT2kveJFK24qyPE/lZWohX0BoySMh2vruUWAc8oUmfXok8Cdq6GFg2shQYbfQWJ5rC2yepi06aILnFJZ2CWrAGPRhuBn9WJdPDtpSPKy5yo1rltYsNCF0ztu++5SGQQiupi4CwTvUyo+cB4Ibcd23+hIQoZoDclIkWDMYH0ZlsjBDym8XVyGhfLVNma3LVyZh8ZVN2gYgYELtwe/6hDEXYOGnHNxks5Qlr0lxPSPvZ+94nEJNmUUShEZkQeDbGo+Axcc3OF8u2qfrEGIcGEoesGscPK/x4pF8rw8ZatwoDiXTnchrdhmxgMpHhNhtYTu4T3AvPHQt5+XtRXiep4s8y68pkuiTR4tKGGtGe5gQnNPCNYikcFQ1qO9HmqWnoE8VChhR3AcjjGit9ZYEyBSOwj0mCxNSfiSeARdCtw2cieepeKyuX0XJkbP1zf51LK98ixevMfRRotG1RuDatn1imYWJBB5z2wXMP0HzfH2JJF5OqmEc5Lp9oh4kYCyLZrja+gkCPJ7SmlCdUdJJx6h1gZJRYEpH+bpC7LxZqy28Fk4L70U5jF2bEep7zYAQPNq+uzoAcyidxlJr1ZC2ZiOEJOPQVX5som2/iBK8r3uoPImKJgtDK7/ck1U2oZSzBmSaFzJ4+IE+3ofQybPD2+c5ldzbZBNQoVEptONJeRm5vY022paGEpZYVIRP4HXa4L2jd7xvfRliHJTwjENb7gzeIUhA5GFfMpQRC68tFAf5aXtv//QaeITYXYJqfoTXxqzJveQw5lbVMVxd5JlrivAosfno0wgV++p39Y40EW3AGoPaWovzbr5heJy80iHi6kqZ4nwy5d2ltsqpk4WhlZ8SU0QCKufP2X5A4MWyVqWmPgssXDP39jlb7zPeXr6FYKxNtqjC8yjfIU7aWj1FEyxzvutNGwgvRaSQcr8uBQYCE+PQtZlHQP6fN5X0IUPzFKQpcKUmnwCSTteh7/m3jRU3B9hpCifqaVtVydsLp6UqPGctTI5riuK6rhnQWxCEm+pF3/KaudAH4Xt6L2pVC0bLSkDz3NbNVwLvzYCW9osowflRGYnzpc9TJudvlvpYVoQIy0pJ+oSqDAlvQkmw7Hn4FqGdzwxKKbd2fmyphYPg7ZsGKRB5mOsJb7vyMNchwKIcZEyf3nHIx8E4dY0D4USWuo8KQJunzKG0xogROt6+ySHUlN+5vL3vdqUu5gC/Yszawmt/z/cpbDN2lDgP2WuVl4Dxyxd5tfWONJH/YIuIoTYXxjzfKLRvbo8fipSpz/6R0Xjk3ZvnT6lSn7ALTDzlJCAssx7tPsiXazbz92jEMdm1UpWQWl7r998oTduGGCztMN7etaJDUR7GU3VBGBfjoJ+gzzjwLtFU0pUXB4TKsYjJv7XOtabyM7Q5SVdKnXLE7yuY57YFNrG1mRTvpI59CuOabVFcDgaXtzamyLC+qyFFBebCGDE0tbQOLxUbhVK+to1Cm2B8GV5HH2/P6EbjUamKofozW4f9AQtwhNnaZ1lWE9rV/RUwEXJ7TKjavJJLIA/tTF4p3yIAwji/gU8o2rbLzmvQBL+LyQfkU5R1+rTZgrBVydH3+oyDqIKHibwYwdcH8nICxJB1lcEoP+IPsPf4AKFm24pKYPgYL2NWq6aAtCRWN1pV17a6UcnVOa4pZK9dM4d51cXIwPctpULMhTSlrQ04uvko/jCbquAleG3vwrh28RTRlUreGd1af4DlvGSuz9jMDMyQ8vsxSV6EIg/r7Uu1eY044RVqoZ2JRYxg8nmEtp19hbSUpG8exhPKm01SXwbZOdEUJDzu6oIDv+CqUSeaSvoIHaMVvfkEqCu8BH3/+AbkKQVFKnX1t8fuNF0KJxpj/CPFa4NoYBglZkQYLs8slehbSuVE8gat2lwI/WOjUKF+245FTURfQ5+UCXBZXVUMUQrexq8WcQTD7PUwI5iw8itd+RVWrY596/b5ktPc+kXnlMnmFUrhKGtIeP3+nfuq79Zye9fm7V2PZe7T4smo8IomtTZJTYh6ouzXtqFnQB4ZTSWadtoMV8A1RSK8NgHq25svXObVcCFC8i5CjWeW5ng2oWctD89bfnU2thn9UOK4ZpcxNZ/SSUaRF2xrv27CXHAKxonC1eaC4Y0dh5B7fXN73p1jIL+Mb1cEiXiNygjjWzPWUW3h8aV+qiSThQkrP4/Fe3eBVwu2Xu6Y5+8scGzcwNvXaqkm1jk20OTt21puYzBZ59q++zkIJINDmHnWLs8IJh5xSBB8p49xIcj5/v01w5WDwRy2c83zS6X8VqAOP30FtTAT5LwULlYTtvVEMCBI3ginaxELhcp3vOnTnBT9/ryxvoa+3o9Sxt4EbXMR3Xyuz/D2JW8hUi3OocsxhLwbo7ZeFM8ZqQNj4vnwVmRxsjBh5SfAlLmNyZY7xm/nN3PHfDkqIqdU1qLklEyIzyq25dIsbZSumqv9apDz8YoMS1tumCP2HkQcRj29DYyL90Mc8qxtpGQOz1MjQ2tQwvRcxot3E5FIX2pQWowFNiK5Wk8Er8V7RYqHcKwBYRbNSV2/ygM+R3RGX8Mw5bVYSEQx20hc3XxheBm6vmB89YCQUXLYVZXI5d06/Vrujn8xT03+BeGoPDxZmLDyG8RmqS8HRQ2P3sztlW3kikKdWm5vMgk/QdZyW2NrgRIaTCF7H28vJNQMFOF3H4Xk7VlqfAAj06cZBZvdNy8OyFMJEEUjQH3y3aj1P/OZzxy8kzyZR681+Xh/bcMMqnq2ULiGqEfLW4XTNfB4sVKz7wrFWOSFcEN+9uE+gDHyXsYIkVabCw4FMRyGt22j0CbyjsM+qRa+yjyb71oVg4Miy56bMWlGcsZCpMcg9IkMZxQzXfnl7zx6ePs8z5MfBuPLG+Ysf8AA8S6W3bLUbT3mJj1W+xnMPiSYiCPCSx6nK08Hhkgu2de4EDrv7j2x5n2FLoxiW3mzCc/juUQj+JecR2iW+kBVgZB5fyVMRqIEkZMxZey6OiApMa/lmoxqLYIIKGc6j6HQft23ixGCcHO0EW7GzzgiGilmXwjHdRySKalWW8oEjDWZJu8qHyWZBhwVYyKSq61B0OCFoyEzbRH1zMIMK388pAmNRgzWL1/5xKLzfLFNU4nx9X2VAApjE0MWPe9Wy0FhKaTBRPL0scy+4/k8g5C4D8MbracEQV7Zp0zI6hO6KH32qduHUSRAjGKflCUYYuMlIimF4pRfPgmMdN5I1La6kUGhXMLptpZfhiNfqdmnRm7cKbwwX6TQN7ePyMv7eqe23D4ISZFPn+oLkA+RTaRaXYuEyLTKCHmSUtSWWEcTm+dWiShxStIJPI3StVp/32eeUcyQ8gvbIff2wqw8VA3LKPTC+JbyILV9goqgwsISvprSEHKhk3v1JcF8JyZATtXVNmtiY7UYUrPPajEeIvaWU/Jr85QBHjJq2W0C1ITxEYpTUBxELXpR5mMgRB4Ujqe1J3+N4c4VrKt+LayPlZpY81oEETDvsVmG8lrfJd25offOzYVeOfL6ftceBDnMVVSiKGFXbi+6iq3HkIglmXZvvA0n4JlqkRw+h7HhSN1/SrD9rJ4yFyNASXj7nK03ablgl5pleCMRgd+/E+4opbQps3IaIXDfvss3hYb4ABFCn1A6J22aDUg1eBYTaBwIXR+EADGKXXvXBXg7Xk8kQkm7QlJGUhphzLqWXfPwkdq0hdMMe16u7MOa50u6GYo+0RBwChF5IRxrSind4wic5/w2QrIJ0SZjUeo7aYK8qnJFKbLGazC20ahWa2gSuZEv50irvBujOCXYfiG0tfIGnEfPvYm+bl4nvH1JsIWIcm+EnvCyTeBY5rzk1VUvBucQGCv8eL8uaw5yQ2lLVztrwATGykCKVktTmsgFqG8HIW8vr+/bsCSK4HU8m/vVkJNQXVUSxn0Y1ty8x5Juc92nryGQh+Bt3j4ISYarjZBsIl9fIGLrMvIqI3nvfy0nD96GM6w5G3/n6RkdxicwZXr7eX513OauJkqAhKPm7UUEyB5EkvXL8qBa/gYGR8mKt+9b8iKkBpfwtJWAArx99I+zvH28fS50mOE+yGvZSmt9GkwitzdevH3bWAVEXBTffguMZQ0xtqKiNgXznLgI44M17+Pt9TXoG4ilzLXUpAk5ciglz1gzRv6ukcu44Ifaej+awMUYHzLSZeRFKVIlKZNSZK0yIr3NN/0oVWmMI/nybqJLBigH5UcaThYmrPw28YicHygQwfDywtiSAlk5penDz3wJRdsGPlh/A6VW3af5RlgmZ/IdHERzcJswsRSFsdJ62mfDBkIn9B5G6OR/KgtRBhtmjX4Qb322swr4HsVGHolMmsjHlvFty+2NiXmOcmVXHo3Ai9/howx9m2kYh9gYU+TVZuhFicH49+m1CJirvO+kZlgCvD2FZ6yRpSWuKJchDq8mQxwhGRNZamwrGUOkqflQkZkM0m+GCD8vI7QTLsfLl1oyWUGhkrzeensvWMvtc7KNt+8TfgOvStjllaKPrlIJoYzUpGasmoiyGmUkdH28GQGSb0e42EU2AkOXM/l9Spg5lDOx8MYw9/zGNnLcrrE1P7wURem7QlGqQSbCUHSV/AII41BKTH0trTMOfQnJHN6bt/fezS7TEuTo9lNgrIX6tSpG3r5cW4MQHAk5Q3zXxtFY2UTUsnTR2DDlyYliwspvcgkX0kf5yMuXwthoa5Xby1nbesxZu9ibjxCUBrMJCsgQUXrK38Ui56RN39ZTBFIsoSV8faIQYxFLRXmPvmv0pRO5gRkGDB4vxLsQWg0qDC0goShYn7HldSkJL9zWVRlgKISrBFzE1cdQgLkLnqVZHm5CahVdcYxwX+TvzZh1GV9RmeiMfNT6QIwzGYr25VpLNNnC6zhPmlwDnWAY/DisaIZR7nJeMwMTVn55Oy+ubl96eQQbj6rHfJFFFunsVouySJ88LCCs55UIHWKpa8AoYLSeWtXFEHSBAlJEhqsP0QYESGktlor2YbcZFDks4k3VY1hvz+jZespYRIcfoWesNEpRMP0HbWvohcHybCkNI9wnpYm+BtdHdvUFr0ngeXuGo1bvR/bmXXF9iNtA9J14vq4dd4OcJBttlZFmS3RJhiL9NBfO9a4l4Fk4T44LzyF6JpeThQkrvxBWGFUK0SzEwdLy9m3tjmByoyzS7AhsA08QYVxXfTxIG8rIqre1swbUvOX0notS9gkxhZeMohBfI03fpaJy2GDyh/X2PJN3I0DCxpxkVSGxc7EwvEZCBQgi4k80V/pBiSaExrgYysvIDJOjRnNMreErwCEIgfs03eQQecR+BM2+kxKMGQ/OQ1uQVEP+g521iFHzFBmTMuDFSpGDdxGZuZa0Id6NoRCVTRYmrPwmkGLnL8cqI5isS7ZttDJfjVTxvej/byuLNOEesTmHUl4XacPbY1EpPsXsk3PnG1/0XULL20ctm5CUyptNCJeRhxRIWjFMfRrkotG8wxhHPVn+aH6UYoXKbdteGT/dizyrVXFdJClIC6Ka0ndBFPCAwbOITmqGPppuPJNIpE+aBWRKGsGw4B66tvuSmolOPQ8lrJGTsZMyo45LKckQw48UdA7FbkYOPjduUS5UXWo6xSlT6pNnU1ow6DwHBl95iUUXHtUQoZxJ6rstNZhM/EFbx1TABPGIooO+q+mw4AyL8IvH6KPAvIpJZ8REMMirLhgv1h6LTon6ljADlFw6IU3ybk2vq5EGb8DrtzWNeAYhcd8uRsbKuBgfhFvfEDxIYfKB0a7tWkQO1OpFQXLlPhFagNFSwSBTDCFla4NUleEUbdUiHamISIAMaciqrdGIhi3zgXPJ4d3JKmUnI+S+FhEamynh+REeBkVpg+eYb775BiU8lt2ksfIYeEbAYYB4YUShQeDtDKy/awoiEHFuHP7mc14VY4oJjTKQ7+TXd7C2Dp8bSBt/yJ/d1/P4LD/f4Ro+V2biyZZddtmB0fD32nM5/F2IJ2fzu/6MRtt94jvOkSvaBUkdnPUvvUvzcE3nMLI8hHejgLx/fk/Xp2jeRyQir/ZZvId/pUkiNJEBY4pj8QzGOr+nwz19RzrCUC2++OKDFml/c9+4bvNwT9djsAm+MZIe6IrLn9f3HXJyyitqNKZCcee1jYvre3+bpArZ5eEqPSWZ8t88O9kkS7pKgzzM3z3GmQzJ/T23DjwKmz+38/2NoRU5MDoqKa7FGHsu5VnGebHFFhsYGdGY7zIicb94NvPGYOPGJgsTVn7h8NJLLz3I6x3CfCGmvFPIxaNQVP/ycP6bwDqXEjuH8IsefNY8fM85GGvEou8J3eVmBDaun5/vX6Gic/08l8nlQeL+zcO9nWNyfMcOQf7mHrXnisP9TBZjwVP5Tu0+cfgOL0tYCKtnEyU136V5+Fxpbplllhk8JxLVuBjr0j1d09gaZ9/J58N/e08NVjx42zP4m3yborgvRfD8vtM2Pu7h2aRAvmfNRsiG583v5b9dy7XnmWeegQF2/fyZm4e/x3gzFu4hyvGdkkw5nyE0X+TU+SFL+bv7b/dmRJxDhnwnH+e4t3+Vrp2ni5RMure/iwDMkc/8K+JxXZ833z+uaX6N8zCk6YxiwsovX2IA1JGFgjrplKkw4s1DKGSADJr/dh6LWzo3P4S2lAsRgh/wPda0dK6/CxlZTpbd/1sXXTo3DkSbyIAnFYW4vr+Vzi0d7tH3XfLDc7W9S/NwnhyZcMs5PWPXu/mOcNbRvI+xNFau0fYMPjOeBNdceM8+72osKZA5l1OLGrqeV+MLGeo6Lw4ciWXfyqghU21zR/k5ESQonqI0/iI+zywnl476vPQ8ZIYRUVoWyeb3prwMAfaep/f9PnNtjJw3TDVjRjFh5R8W0oO+TH5A3ib8K/UPzCxo7ulig2cHGIuuhTyzAnLWWkNWDfgWgp9XHmY2hNzq8H3njmERlreB4mHo25Y7g9QFqVsjj4clbv9ZmDTlH2GEEWYv/NOUH7PLkyFH1Hr1BvQpM43QD7fd9Mt08fnnpDNO/5904gmfG3iqz33htPSN7/wg/ezaP6RZ3z82ZwDBh4TF6CME9YP0qRLNDpilyq+kpVMNo6mFUa4pd1TOkBPJwxA8yCfkiLxvhBnDQ3f9Jp155A7pFSsul1Z88Xppsy3ekaZtt0PaYft3pi032zC96LlLpmevtG5678mXpL+NLMAMQypBdh1IXPm+6oYeFPwMTgA3JpVQSu7TVTpZmGHll4vKgRBLSi5aJC1K0R6qyw2LiVVW2opBah6qBuqfyh8660bH9EPuWNoIog0P/e2X6ZRD90p77Ht4+uZVjXz4zmvTiXttlBYy7i/eN111W/fCpBzq3p6p9Kxz6mE8NCIp9ZVk26G9XXVBhQZBaf2DMq0mOP0yDIjot+9CqJmFGVZ+C3f0gttTP8oowx5KbUogFjUok4yO6Ye2XLXxvh1u0/FguvWqC9KJR3847bXztuktb9wkvW7MG238+jemLbbdKW29ydppsbExf+zqe6af/bnfNldAMHWfeabSs86phzIebx9l2GEP/THKgUsttVQnITmzMcPKj8HHpFqQwONr0GDhGARKbVNCXX9qvaWXd6jRKgnpMpM/jY7ph9KPqGoYj3D1qe9JL1z0ielJS62Zdv/46emCH/04/fzqa9I111yVfnbFpen4fTZJi46N+WNfvHf6+W39ttMC6yNEZkLY0rPOqQeZVSJkFEuy7dD8Rgc4SL0uDIW+B62+Fl6JlpUfpQWTiZme8xMSeY3ynHxfZGB9vrqpGisrabGGfnQ1fAbCr+waJGnDCDOGn564U1pu4fnSvE9fPe1yzCnpW98+L53/vQvSed84I/33Efult7x8+bQgoXzqxunMa4dLKUZ4NBDW0dCj4YgH1+Og+1XzjpZvHZH4LN2OVkpKnxj0Pqs9ZyVmKeHXBqyoVkvNDXIfy1HVbUeYUdyZrvruaenIA/ZM2229Rdpiy63SNtO2TTvtvm868rNnpkt+eF467uB3pa3fsV/6+i8nr6HkXxVagRF8WtzxXnobJtuDTxT/NOUfYYQR/rkYKf8II8yhGCn/CCPMoRgp/wgjzKEYKf8II8yhGCn/CCPMoRgp/wgjzKEYKf8II8yhGCn/CCPMoRgp/wgjzKEYKf8II8yhGCn/CCPMoRgp/wgjzJFI6f8BV97PknKctRcAAAAASUVORK5CYII=)
A square plate of edge a/2 is cut out from a uniform square plate of edge ' a ' as shown in figure. The mass of the remaining portion is M. The moment of inertia of the shaded portion about an axis passing through ' O ' (centre of the square of side (A) and perpendicular to plane of the plate is
![](data:image/png;base64,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)
physics-General
physics-
The fig shows a uniform rod lying along the x-axis. The locus of all the points lying on the xy-plane, about which the moment of inertia of the rod is same as that about O is :
![](data:image/png;base64,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)
The fig shows a uniform rod lying along the x-axis. The locus of all the points lying on the xy-plane, about which the moment of inertia of the rod is same as that about O is :
![](data:image/png;base64,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)
physics-General
physics-
A thin rod of length 4l, mass 4 m is bent at the points as shown in the fig. What is the moment of inertia of the rod about the axis passing point O& perpendicular to the plane of the paper.
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)
A thin rod of length 4l, mass 4 m is bent at the points as shown in the fig. What is the moment of inertia of the rod about the axis passing point O& perpendicular to the plane of the paper.
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lTiFjnwMOgpgvF0igCbV68BGa1PV4goUPQMAFAYCaTzVCBxMOELL3Toroq1ntKxbFrFsOEbmt2p5DwYqxUkNHzy3Uo69gb0IN5vmEZbf4dQnq7a4ZNTlTw0ksveSvpD06APbLNhsQkBdpmoT0JSi6pYrRbWLidihRbowhkPZmvyC1kPjxoPEHax4KKuPFyCLOpgQwbT5aLrRN190VnqX6n8D7ddMXPYpoTsdHywvqO0QHDEh4FgQQN9XJs6CBoA6QqwQ4HMZZkss1IIl1BxiACYkhGGDwv+zMCkjFo9EGRVSpTrI0i8LDxTGzPgHs98grve0AfI8TjSsd4bBr3gi44s7K6flO8Uixk+dkwZhaTxAplMeYhZEIdIb3QhZm9JEwnUxo08IMDkCQb36ft+WMICMHTbcPbUFmBZ4wWGwT7gAPCHsPHQceBl473axd7owgUnLIx0JP8MACPkEZ4PMkwAZmWM14MOjGC2a0RXW7S2RferfG/rNLGzOJRts1n52cxzRpQiuIZDBxk9DBCJMDsrObeQuIMPZPvhe/Lhu8PYxx01QGZbCZ+B/29dID3loM9XsTDNaV0gZkgOVC2E3a4pTrOKHpQaoBHQR2bmfFE9KeMxB5x70PGEsMYNKZK2YvUucHpOr5iY731/qeanlE82sSCsphk8KlvCxrhRvYSb8S8pa8w4fsgGdC9e3dvJTcYRiZ22yEi3iW9v2YPfbrA4BMioaFDh3orueEzQUJCejD3A+tMlUJbT1ecUTRg81BITAhnZ0V5eQjFgmoVEfPZ1GS1c5G9Qu93vkv1bn5YH/8YXAeZblADyCFhZjG5m4NsLwePDUaSxBWJl30JBplnFNYKiKdIOQ4JGBM/fEyXchPggKfSwp5sY0IhPB623bTAyDYOmHSeP+qMogWnIvcNU8dmbmIypazxogR1TGAE0KDsUGP3jiztLCaVOH4W0yx1wdOIVx5DqxghdYEL45MAbxCDHdYKSIKHshPbEPC9I6+YdwEVKXZkamX0e9u4LbEXjUQhbY1mJtmEAQ9BeiGHv33gpSPOKAaADoXOhG5i6i1kK9HKbMPnQ7jBKYoWVZjJgjCyNm/SujWrtWbdBm3ZHn9D7MrarA1r10Q94LXauDn8e9u6fpmmTRqvCZOna9nGxH8GDAk1nnQNmbCx8njQHgwdoFwnrC1vX0DhdljGFCNB/zUlQ2aJCT8b4eK//vWvAuqLu7Rs2ufq1uYuNahbR/XqX6+Gd7dWr4+/0Zqg8seda/T1B911X8PrdU2dq1W33vW694keGjN3T+9+jF1b9N3gLrrlqlpRb7aWal5eX4++8ZXWxjmF0XXxiOMN56U0xx7oQCKKwy2oHTPdcEYxDoR06C5mYzweDZuGl4TQyoYNTmiCRxmknxUGWzKma2DXVrqiaiWVK3+Ozqt6gWpe20QvvjdOK23pc+cGTR7yshpdfaEqVawU/a+Czql2pR7q9r7m5CpI3q7FY3tH/9wlqnn1Nap39aWqcU0zDfx2eUJTfchi4kn7Qzc4FAi1wjafX64Tdk/3vsRvBQzzlHjuFCmbBpCfi8x5vJ7gQHZmaszrLVT+6IN01EnldE3De9Xm3w+pUf2LVfrYo3TOje01OWPPQ9uZOU/dm16ikocdpwq1btIDj7TR/Y0aqMJfjtahf6uhbl8skG+ul094RTX+cZ7aDvw+lvyY8/mLuqzsuWo9eEbg8ApKjUis4DGHQfLQHgTBO59f2U464YxiPvh3Sdsvkq+dBWlNGEOMIjV3hX3FQeb84Wp+UWlFIkep6o2t9MJrffVWr5fV+taLdGyJw3Vhkx6a96u/JdZpeNdbdGIkotNq3Klne/RWvzd7qO0dNXR8dO3s+p01c70Xtm6eon/VOEuXtRmsHFl9o/reU01lrnpCi/PpbsOrQHfzvWrEePRXtMKgjhU/exnmQe5rkEg4uML6qumDJhNuZ1cpS+FdSHwIcbaWjemuKsdEdPDfauvNr81x/pka1aORTog+h3Pv7aOVsfNpm8a8cpuOia6d0/xNLTfOrGXje6raHyM6ulJjjVqe48H/MqKDzil9nrp8meP1bpjaX7XLnKGmb33/m+H04Z0kkRhv7iSyB06APZ3eL9tJtuUyVXFGMQHIQDPays4wU0bCJgkasYSYTXsbWdjCu+Jgld5qfn7UIB6gix98V6tyuXDrNaBVDR0UKakbX80pll77TU9VOTKiwys006Rckdc6DXj4Yh0QOUI3vzpRsS22ZZaeblBZVW/vom8XrNHK+RPUrsE5qnL7S1oRp8ONEhr74imMBsXtQT83XgdhKH3k+0NHDINWQPTPMK2TZ4r2ZuuLGHI8xsRC/kwN61Q7+rwiqv3c6Lze26YZanXuwYocc6U+mh/9kLfOVpsLD1XkyAs0YLodZWzTkJYVo1/reD3w9g8xo5eduUJDut6r2jUvV/3rGqh2rct126O9NWdt3gfGdRsUqod95sgJHFT2DYj+TYtB13ikK84oJggvC56h3eOJwcSDCJrKgi5Fhg8tJkzcT4pNY9Wk8pGKHFZRvX7KqxttGt9DFUtGdORFHbQyuom+ePpa/SG6Iet03dNR4rNh8luqelREh17cTqti33q2Ns4bqaeb36Arol7f5ZdfoZtbPq8Ji8MHYBAiI9ibI/1ZQ4gPyjQD8ysxKvtKWkgGvwA5bMo6PwN3/eD5+vBM0ZXRGPPXFzfpk6dqxoziDT3zPgP9+rNaX1Qi+vunq8ukFcpc+rnqRL3BAyrdpykZ9tferR/7NVaJ6Neq3vbD6LHms01LfpqiMdGfYdK0edoUoHWgA3JIhSWKGIyBHGRPysYQ0hFUWMOWUwVnFJMADwgDx/0T5kbBm2DzkIAJOokpb+C03euyjszJalGtlCKHnKWXvsvrDWQM76wzD43ooNMaa3LmEr3RtFJ0w/1RrT8JuIVtwZe6uewBipzQQCvWGAmVnVu1bvUqrVqzXtvyERMxhmRr/RpOjATZZD6fIJAcCLOLUvYSbx9PN2h0PuEiEgiTlUyPkfCfBNGAAQO8lTCyNf/zTjrz4IhKnn+3Pp9tuuuZmvDWgzrloEj0Gf1N7UbM18o57+rC6K9LXvaYZgRUvPw85CEdFzWKFe7pp+VBomEA1F7G025JHqIF21cyJFK2k5ts7ciKE1KkEM4oJgkhIRlmNr+ZYUZPo9CX5vigE5nQmxDcfvmSI1NfdW6gw6Mb44xr2unruRu89ehp//MXannl6TGvpOQZDTVu+Sy9dEuZ2IbrOCZA11w6Wo3PjXopJWspY1Xy49Co27SzmLQ+EoIFZXb9yTXxhg78XnCYhbUC8lzxlqi1NJk0aVLM2AQl23KRtUJDn7lFf48eVoeddrGatemgZzp31AO31tEF51ZUpXJ/iT6jU9Rx3CKtnjfoN6M4PcQoHht9vpXu668VCRhFjDezQMm4B0FrH3ff8M7akQ7SgXnghbF9w1JNGvqaHn7kUQ0YnR4jT5xRLAAYPTwIKvvN6zUp/ibBgmcYBH3UaJAkbwrMpgXq88i1OjG6eY4/s5quvelm3XTDNap6bhXVqVdblUofrSPLNNaUjfPVvWHZ6IY7Se1HBhjFJSN1R8WDFDn6KmWszieTYkEhNJ6v3d6I3hYkE5DlpVwHXcvW6IoClGBhHNA6zQjAB8+W0iI7XEYK4IDE24pPlmaNHaT29zfU5TUu1iU16+iuti9rxISv9epd/1DkoHPVZ+ZGbV8+QtceF9GBFe/R5Az7c9qtqb1v08FRo1jryY+U379IxIJuy88UpiOSBMNg2vooBe4cBGESiMmOTSs1pks9HVHqH+oyLj0SMc4oFhA0Mfpi0dQ4kX0wALxMhIpBmwUBnxcOr6pgNW+wRfMnf6bu7f+lpo2b6v42nTRwzCyt/u4d1fr7wTqyWnutzl6vgQ9VjxrFQ9RsYN4pMdt+Gqo6f4mGbmfcpYwN+cTJBngXzEE07+HAa+rTp4/3q7yQ8cQTK8iA3v0FJTdcjhXWCsjPh/dkgr7IIWjfVWKyde1CjfvsE01akPOzZ23J1OZtXm547SjdEH0GB1W6V1PWRd+FbXPUruaRivzhfPWear87mXrnrrOiz/NEPfze9MCSGxMOKLzEMINNUf0rr7ySp56WJgUkjmQunpr8UkNVvLSFpgRXOKUczijuBXgYbAg2kz2eijo3Jkeb95H4UC9GATEF4kGTvsPJ1tyx7+vFF3to3KK8BvWb7rfpKDyJp3MyiD+9/UBO6U3Td3IyzAazBj+ik6O/d3qTvtqYRBKYnwuvz/eoCJ/J4obpa5QymeU6RRnfAw5KLJBl5+e0Q1G/qDlMFtgw80NdfdKBOvDshvpszp7k0qblU9W1YbmokTtWzd6cpBxffZsm971bf4w+l7Nu7qKpy70oZMdGTf+4oyr8IaIjq96vCcviH6ZMbiK0Dxr3BRhK2i5tI4+Gir6Y1CzJ7CXqWL+SajzwntJDUXRGca/B2yOxwMYwb5xDi2EkPvVuQTfRscmo46OlMBkPakb/e3R0dNOcVLO1xs1epR3Zu6PeyAKN6v+ozo2GXqUqNtXo5V44vOpbtbr4+KjncZru7/WVFq7bqh1b12n6qN66vsIRihx/nnpOzIia2sTwL55CH/RBiMe4B4HHjO4Y7+rRooZ/i2OQVoiXH/Sz+r3vHHZ5yF6n0b1aqvwJB+qI0ufrxsbN1bzRjapW5jgddtw/dGeXIVpmWpNtSzXoyetV+tADdOzZNdTwrua6vUF1nXDIATq5yk3qPykgaWbglxqR+AuDAwy5wNQRkTX8i6fy0xFNshcMUt3y5fTg4MXeSurjjGIhQfcGG8POMFMOgc4WFCqj6TRp0iT2IibcYJ+VoY+fa6yyx5fUCWdU0aXRDVz9/LN1/DHHqvL1j2jY7NxfZ+PcL/Rg3fI64vBSKl+thmrVqKbTjj9cx5xVW12Gzoz6JonBz2CXr+Al4ZXYIRiwsfBGEs9eFh3wBskuM9nHBn0OI28XcBOK4mUFH3A7tPC7L9Xrucd1b5M7dGfT+9S+a299OWX+b96V/35krFqnbRuWavrYj/Vyx9Zq1ugONW3RWt36DNG0X4JnHvrwHKhFJHoJ0kb5N4hc0EjtkigKtznYk202mDPgQZUrV1dDl6aLn+iMYqHC6UwixR6lz8vIMNOgDg5CFsRwdDrTA4tPllbM/V6fDHxdXZ/topd7DdSIb2drjVUquWrVak3+4XtlbVimmRM/V98e3dTl+e565+MxmrdiT4IoP0ggIRGYI7TwfkkyhBWmE5pR0pGM11FUwGuK1wqIp4X8Yc5+RF/kcMv/9sHsQM+ckJX2QoZjzJprlAcFHKZhYJjRbsM6T6iA4GeyB8piJDnwki8ZW6++912s8vWe0/Lfrw6/0HFGsZAZNmxYzDDa4QvZS/Qqir9tz4oNxWZio9mjqwoKhphe1dtvv0M7diW+sWz4OhQqE275kH2n9IZNGOSR+EMHzKtkUw2MOUm0p556ylvZg2800VdNMJLICwwnTpaRI0eqRIkSsfKYgsAsytAQPgqhP+/l8OHDY8/UB32RrqsCXTy1ZrzuqXaaGnQbH/WF0wdnFPcBhJT0kBJSmxBy4T0FFQoDRoa/Vxgj3glz6RwhlNqbO0+CLp7CcOM1BhlEspcY/301MHZ/QviMPseQBBvCZPRHO9wkscQzTPZwo7j65JNPjkUMyXb7EBJjjON1nqB72m2qPD8mkmOIk0v45bBqzIs6/9Tz1P2b9JoV6oziPoITm9YqNDWzNo+aMLREjFbQvRhsQE78sLthEoHTH32LMI/QF6NWkJFPtMHxvZgXT2How1oW+XkI3zDu6YI/BMLuCQaeIwOJyVqbcAhRx5roAAXfW0PXo4UQLzRRiDrwaDt16uSt5IVnjydqw8QfPH7ekeTZqq+eu1ZnXNRS3yzepqxdRa/+tKA4o7gPIftKhpnw09TW/NCWRESQjsi1q2g8ZHsLArV2dCMQlgNaEcYtrEQjCDxDwnl/UgxfixALAxv0PfMzsZnRxPx/N13AWFFbant/eFdMA0JbNQugWbd7wuPBwUnNK2BoOUzDBlXY8Hd5j4J673km6Ns8e0a1mfhlO+aBlxRb5+uZq/6qU+o+rH4ffa3561NPOw7DGcUE2Z21VVkFOAw5hfEmqO0zM8yEX+hVGJmgcJrrNXlpw7pjwvDH7ttlQIS7JEvMDpwwyLDSrcOdvr7+RChJ6E+hdhB06RBqBtVlpgMUq+N92949BwDaoj0pibpVwmg05ngglaC/mnWciRoswuWgUV8+vGMYW/twxfDy3iVqeAPZvFi9WtRU1auiX//7DO0ydMpUxxnFeGRv14qfx6pnh0fU8rE3NS+f6dZhsJEIi5hJZ5d5kJAJC0fRIPFQwiZ927BB8VCCwlfCLLw4vLn8Wu3IlOP9+GEVXi7eZlCJCuCBYoh/j4Gx+ws+Ww4KWueCPj+SLLamh0HEaIUN2sW7404g5hXa+Fpu2GQbvFa027DOEw4/3ivKpszECjoiXTht27b1VgpKtrZuWq8NmemUYsnBGcV4ZO/Uul++1O2VjouGCS+KTqyCglHCIKHh2Cc7xifoUnbAi+TvUFibX1iKt8ZQijChnhAPby7w5kEPBh/gpcyYMcNbkbp27Ro6dZnvnYQOfcDpDp4+h0XQoYPxCToY/MEKQfWLHIh470HaMoaM60V5L+yEFl+LAzbImPrwvgS1XhIxxHtHHM4o5s+GcWp2UTk1eiNvV0qy8HIjwlMka2o81MLxsrKpmKJjN/Cj4ZGVxNsM2lzgt9vll/X1uy+C2u7IpNota/40HLLKNmxcOnK4FjY/7zNd4LDAQ0P3tUHq8LVBH54tz8729vEA+ayDEiA+JGBIXJltd3zmhOvURIZljHkH+Nr2QAf+LQ68ZLTl4ogzivmQ8VknVTnrEvWZGb+bIBk4wTFMGCgTOiXIWgaFqQwsIPylA8YsjwEMEskcJn0nAokYWx8jRA66h4NES1hY7A8dQKMqThCyBhkXss1kqxmdZhosf8iCWVFA1JDI8/KNsD+RiAMLg2cWjttgsG1ji9bL34sXJThycEYxLlv1vyev1lmXttGswrOJMShwptTD7gsmHMXocdrbngBeB1oQ4Y85gILaOF74RFu0MIAkfsziYxIJeCX+xVOUFNHuFuaN+PME7axmcYGDjex8kOaH3GDfheJ73ITgGE7qChMZzQXcSc3hg0EjM20fpj58L5RM2YcUzxDt0i42dwTjjGI8Ns/QY1eeqZqPfqJ9MSnOzzDbxcHojxTUoinZRgldkVIPRHhCYP4smmOyXRSE5BhSkgHoiGbBMZ4OWheDLoKMIp5sMmUj6QrzITE29mfke3f2gefri2Tx85/anRtC6BNPPDG0BhQvla4je8YncOARwgeV7Tjy4oxiHH6dNkC1ziindp+Fhyp7i+818NKbuhyGiRAXA2gL7ehKaJB4KtSoEfKaGcZEweOoUaOGqlWrliuLSdIE/TJoOClGmewlBru46IhhYIgwcpRW2Z8/nr45cxKQK/D4eG4cZsnAMy5dunTMEAf1YxNe8y7YCRQ/A24XmDvCcUYxlN36vm8TnV7+Fn2xeN8WI6M54e3x4pubBY+MGjPCoaBNRKdCqVKlAtvQEgEDV7lyZZUpU+a3zDabh0xqWEKH4bgUpAdlTIsjyBh43P369fNW9kB4PGjQoFwGk5FkHIJBHTJh0HnCAYlkwTg6u8QHD581X/rwYVgHBtEeUOKIjzOKYexcpTfuOEeVbu+uZXlbfAsdhHBefDK5tsHh98hqmmUyPv4QVzZfshAeE8rhYZB8IZwn2RJmENnIYZnr4ox/sb9t6PwMs32BPgddvNv1TPwbB81CbtNDR/slNLYNJV4sSbuwS8Qc4TijGML25aN0yzlnq9nro7V55/ZoCLsXRYoJwlWTjHbiP79wGtCCqEnzi3XtsJXsML9H0XWi+MI/2iLiPy186Ihhw0nZdPz5grYepjtounhl9ig1jBmG0e4wIvuMBhhP9uDZoB0HeaGAd8/XYPCt7+n74FFSyG+2lzoSwxnFEJaPeVKnHldWd3fsp3E/zNGv+6mdFwPI5BIywXZpDhuPKw6Ceo/ZdHgfiPlhFxX54F1gRBkj5UNBOQYxSIxHw6JVMWzCtiMHP4Nve/p4+IxRMxMg1CByCIWN7PINHmU7YdotnibPzK464D3hAEs0u+3IjTOKIaz6tpeuvfAS3f38UC3bvDvhkf2FARuCqScI8nb3C0I6nh2hrm38/CsOqFkMyzQSVhGmm1lMtMmwC+GBoQcYY5e9jA+6L9lfJBA7I02ITXeK6bnRhoccEVQHSmkNpVdhveokVoJqDv3e93hF4Y74OKMYQvbuHcrcuEn7QU4MhaQGIZk9GAAvAy2QxIxtGKlxpMCbDWgOoPCh/Yvf88MtvAq8xjCdkBo5+q9tzcoRDJ8/Y8DsOkV0WjQ++7ZAkmR2vaOv3YYNjGWIBDWu6MkmyC8Y0nQa3fZ74IxiEYd6NjaI3fjv1xJSKGzrUngX1DkiwJshOMkYWgH98BujSbhnby4ff+iAfb+zIz6Eyhxm9oAIBnzYZTocTsglXEkBhLxkp+Npt3Q22bcKEmL7o9uCSnYcieOMYgrA5iIDaYdLlM5QIIxxs0Nbwjc8SUp9+HO0pJkZUsp8GDYR1gXDxmLoQKITehy58fuM7QlIfK50M5l1g3j+eHjUhxJ+x9NuKQHi79o6IxIIB559sb0jeZxRTBHQ/PAYbQ8B8Ewo2bGHoAJhHKI7XqU/mxFdEl2RqTpBsOFoCaNbw+mIBYfe8KDSG0Jos6USSJSVLFkyVtwd1EUEJMbQjO2ZldQ+hmmTjuRxRjGFwOvAwNl3riDes0bIRgLGhrCZQQKE2YRrzATEiIaVa+CRJtOb6wiGTiRCWjReElw+GD3kDfP6VzLUp556akyPRBu0wTtEyqCUygSPn9FtQYelo2A4o5hikPBgE9Du5SdLfAih7BpHGwT/oKEBPngseB30Qzv2HrMV0ARvnQ4VQmfAm0RzJHPN0A/72ZK95loEEw45RrfRkmn/eUfBcUYxBUEHJJMZ1PyPsSOcCroRkL+H0B+UlQbq6xiiat9C6Ng7SIqh99ntmLR3Eg6bdaeExhTh++PhkC/QdYOuiyU8RzO2R8k59g5nFFMUPAyKeynNsTUrf/qOmZjBeJKZpBg4qLibUI87WTC0YZqWo+D4rYDm2C+eAx4/bZZ2yIwXSFKGZBnht+9R+vij2wrjOlxHbpxRTGEoFuZeZ5Iodh0h4S8aFBlNNhjeCK2CeC1BMEyWejmXvdx3DB48OKb7mgkxDiu0xaAkGeE0nqSt7XIIksAJS5Q59g5nFFMcPDya/umRnTZtmreaA+I91xugN4W1igE30eF12DfSOQofEmJknk3dl2dDyZRZm4j2y/OwM81Akgzd0a5PdRQOziimCehQeIZBCZJ4mwc9Cg+xoOPHHMnBIUbmmWJtc06mP4KMqwQIq2nVtLtioEePHrGaxqAMtaNwcEYxjSA7edlll8W8w0TAg0RDTPRuF0fhQH1iUDseLZUM3uCAYsKNPRCEgnAqA0ikOfYdziimGbTsYRjDRoCZ+Bfk28NJHfseNET0RfvKAgaAkIyxtV/CaGpUk712wpE8ziimIRMnToyF0hi9MMhQ43WEXcTv2PdwgCWSQSacxoPs0KGDt+LYlzijmKaQZCH5wrBRuwSHXmj0K7sY2LH/ofWS5xRWFQAM/WB0m9kV49h3OKOYxjBfkUwnmqHf0kcZDz3NdoeF4/eDWkU6kYLaLtGHCbPtuZqOfYczimmOP1+RQm88DWY0MlIs7B4Wx/6HZ0EroB0e08WCDOIuntq/OKNYDKBAmBKQiy66yA2MLaKQaTZbAWnvo5WTqeeO/YszisUEQumqVavGDGPQHS+O3x+KtfEMuTMaHZGQ2g162P84o1hMwFtEl6Ltjwv4HUUTBgpXqVIlVlZl1yk69g/OKDocRQz60PEWHb8Pzig6HA6HgTOKDofDYeCMosPhcBg4o+hwOBwGzig6HA7Hb0j/D2kZztgw//WYAAAAAElFTkSuQmCC)
physics-General
physics-
Figure below shows the variation of the moment of inertia of a uniform rod about an axis normal to its length with the distance of the axis from the end of the rod. The moment of inertia of the rod about an axis passing through its centre and perpendicular to the its length is :
![](data:image/png;base64,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)
Figure below shows the variation of the moment of inertia of a uniform rod about an axis normal to its length with the distance of the axis from the end of the rod. The moment of inertia of the rod about an axis passing through its centre and perpendicular to the its length is :
![](data:image/png;base64,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)
physics-General
physics-
Two thin rods of mass
and length l each are joined to form l shape as shown. The moment of inertia of rods about an axis passing through free end (O) of a rod and perpendicular to both the rods is:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANMAAACfCAYAAABjnezOAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAFxEAABcRAcom8z8AAAyESURBVHhe7dwNUFVlHsdxSrRkzHEszQZnYzQ3ZN0soXxXlNQwYzVnWytXY9nVbUdzJV1zdWtmG6BJsdFaRZx2NbC0cQZ1y7dKfMHxBfGVTFBXUVBCM/AFFC/3t/5PDyovPvLce890rvw+M8/Efe6BsXv53nPuc84lAETkE4yJyEcYE5GPMCYiH2FMdAsXLpw7jbzcgzhRckHNUUMxJrpFBfat+wh9u/TER5tOqDlqKMZENZzemoyuT8Yg81SlmqGGYkw+cvHiRZw4cQLr16/H66+/jsTERHWPf/kq4WX0G5uEH9xq4i5TXFyMvXv34vz587h27Zqa9Q3G5IHS0lLs27cPK1euxOzZszFmzBh07doVDzzwAO69914EBATgjTfeUFv7k2JMiemFP8/brG7ffVasWGE9P8HBwXjhhRcwY8YMpKWlYevWrSgsLFRbeYYxabhcLiucQ4cOYfny5Zg+fTqGDx+OiIgItG3b1npSao/AwEAkJCSon+BfXMdWYXD3Pvj3nktq5u7jdrsxc+bMOs9bUFAQQkNDMXjwYOvIIiUlBdu3b0dJSQkqKirUd+sxplrkMEAeyGnTpmHo0KF49NFH0bx58zoP/u1Gly5d8Omnn1rxyX9/7rF06VKkp6dbh6B3si/9LTwdGYcjHh79yKHuuXPnrEMopw55cSwvL8fkyZPRrFmzep/D6nHfffehTZs26N+/PyZOnIgFCxZg9erV2Lx5s/WzamNMtWRnZ1uvUrKHqe8BvtN44oknEBMTg4EDB2LAgAE/+5BfBBmZmZnq//B2LmPBXwbgufjl6rYZOUSKjo5Gv379EBkZ6egxaNAgPP/880YvkrcOOZSPjY1FZWXNRRrGVEtVVZX1Cit7qE2bNmHevHkYP368tfvv1KkTmjRpUu8DXD1kT5aVlWX9LHmwnTLk8Ean6oe9GNM7HP9cmw93lUvNNtyVK1ewZcsWrFu3Dhs2bHD0kMO3qVOn4v7776/3Oaw9WrVqhW7dumHEiBGYMGEC3n77bet3o/ZjypgaSCI7fvy49SAuWbLEejIkMHkj26JFixoPfrt27bBr1y71nf7hdFYyfvmLCLyfvhZHzty975mEHKY98sgjNZ4zGXI0IuF07twZo0aNQlJSEjIyMrB7926cPXtWffftMSYvyNKqHIPn5ORg8eLF1nG4vHo99NBD1iKFPzmzOw0jo0fio9V7cfXaXboufp2swlbHIy+EvXv3xiuvvIJ3330Xq1atwrFjx6z3VJ5gTDY4fPgwtm3bpm75Czdcrir19d1LFmJkgWnt2rX47rvvcOmS7/bCfhPTsmXLrJNtRE7lFzFdvXrVWsaUhQBf+eabb6yTdbJ0zNG4RvUpA2/HZ599Zu3l5P2zLDr5RUwHDhyw3hyGhYV5vVuWlbqNGzdaS5uyPCrLuRwcngy5giI8PNw6HyVB+UVMycnJ1hvGe+65x+tVsoMHD1qX/sgbTbnCob5lZA6Ohgz5/ZFTAbKIIS/Sjo9J/sFyCU/1CoysunhD9nJyYlWWuYm8JadK5KS4X8R09OhRtG/f/kZMclb/TicgdWTP9NRTT1l7JiJvyXtvueqjrKzM+THJNW7VIcmQa6Vyc3PVveYYE/mSX8UUFxdXIyYZqamp6l5zjIl8yW9iunDhAjp06FAnpldffdW6vMcTjIl8yW9ikrPUTZs2rROTXPsmF6J6gjGRL/lNTG+++WadkKqHXEflCcZEviTnLCUmOYpybEyy1Cjr9/WFJGPcuHFqSzPexiQfNSCqJtdgymek5BSOY2OSy95v/WhD7Q/ryYlXCc6UNzHJZ2HkYxfy9x2mTJli7Tk5Gu+QT2OPHDkSDz/8MD788ENnxiTnkeTvKEg08gEu+UBWVFSUtbInl8tXB7VmzRr1HQ3nTUyyTN+6dWuMHj0a8fHxmDRpEkcjHvI78NJLL1kfCJ0/f74zY5JDqQcffNC6Fk/2BuLFF1+0LioUixYtQkhICObOnWvdNuFNTHLJfp8+fXj1BN0gf2xFjqKEY/dM8jcLioqK1AysiwrnzJmjbsGK4cyZM+pWw3kTk3xvr169kJ+fr2aIbnLse6baasfkKV/ElJeXp2aIbmJMBhgT6TAmA4yJdBiTAcZEOozJAGMiHcZkgDGRDmMywJhIhzEZYEykw5gMMCbSYUwGGBPpMCYDjIl0GJMBxkQ6jMkAYyIdxmSAMZEOYzLAmEiHMRlgTKTDmAwwJtJhTAYYE+kwJgOMiXQYkwHGRDqMyQBjIh3GZIAxkQ5jMsCYSIcxGWBMpMOYDDAm0mFMBhgT6TAmA4yJdBiTAcZEOozJAGMiHcZkgDGRDmMywJhIhzEZYEykw5gMMCbSYUwGGBPpMCYDjIl0/CamYcOGITk5Wd3yHGMiu/hNTEOGDMGsWbPULc8xJrKL38T05ZdfIicnR93yHGMiu/hNTL7CmMgujMkAYyIdP4jJjbLvT2BH5lfYnlsAl5r1FGMiu/hBTFU4nbsSz/2qMyambrmelncYE9nFLw7zXCdXI+rp/vjPzlI14znGRHbxi5gOpb+FHkPG4XClmvACYyK7+EFMFZg/IRoxk9O8PsQTjIns4vyYftyH2Mhu+Mfq42rCO4yJ7OL4mM5sS0GPbkOxqfiamvEOYyK7OD6m9e/9Fs/8Lgll6ra3GBPZxdkxub7HW8N6Yfy/NuDKNR+sPlzHmMgujo7pWuFKPBPSEXHvfIz9J0rUrHcYE9nF0TG5zu7GOxPGYfaybSh3+2ItjzGRfRz/nsnXGBPZhTEZYEykw5gMMCbSYUwGGBPpOD6m7OxsFBQUqFveY0xkF0fH5Ha7rb/9sHDhQjXjPcZEdnF8TIMGDWJM5Bf8Ys+UmpqqZrzHmMgujMkAYyIdxmSAMZEOYzLAmEiHMRlgTKTDmAwwJtJhTAYYE+kwJgOMiXQYkwHGRDqMyQBjIh3GZIAxkQ5jMsCYSIcxGWBMpMOYDDAm0mFMBhgT6TAmA4yJdBiTAcZEOozJAGMiHcZkgDGRDmMywJhIhzEZYEykw5gMMCbSYUwGGBPpMCYDjIl0GJMBxkQ6jMkAYyIdxmSAMZEOYzLAmEiHMRlgTKTDmAwwJtJhTAYYE+kwJgOMiXQYkwHGRDqMyQBjIh3GZIAxkY6jYpJ4vv76a+Tm5qoZWDEtWrRI3QIyMjJq3G+KMZFdHBWTy+VCly5dEBwcjJSUFFRUVCAmJgaffPIJCgsLERsbi4CAAMyZM0d9hznGRHZx3GFeUlKSFYyMgQMHIiIiwto7Pf7449Zc27ZtsX//frW1OcZEdnFcTDt27EBQUNCNoGqPqKgoVFVVqa3NMSayi+NiKisrQ8+ePesNSUZiYqLa0jOMieziuJhEfHx8vSE1adIEe/bsUVt5hjGRXRwZ05o1a9C0adM6Mcn7p/LycrWVZxgT2cWRMcmhXkhISJ2Ypk+frrbwHGMiuzgyJhEXF1cjpMDAQHzxxRfqXs8xJrKLY2P6/PPPa8QUGhqKkpISda/nGBPZxbExyS97+/btb8Q0evRodY93GBPZxbExydUQw4cPvxHT0qVL1T3eYUxkF8fGJOSyIQmpZcuWOHnypJr1DmMiuzg6JrmgVWKKjo629lS+wJjILo6OqbKyEu3atUNCQoKa8d63336L8PBwFBUVqZmGy8/PR58+fVBQUKBmiG5ydExi586dPjvEE3Lt32OPPWatFsrXWVlZDRqy7eLFi9GpUyekp6dbUfpqb0l3B8fH5GtpaWnWOSvZO0VGRlp7moaMvn37WtcMhoWFoUePHhg7dixKS0vVTyVqhDEdPnzYuvJ8165dKC4utg73GjpOnz5tHeLJ1+fPn1c/kegnjS4mOTRjCGSHRhcTkV0YE5GPMCYiH2FMRD7CmIh8hDER+QhjukXl5R9x5GA2Nm7MRPaBIygtr1T3EN0ZYxLlxVjxQTyeCe2AsPDeeHbws+gdHoaOnbtjytwMlFSo7Yg0GJO7FEunPYcmAYEY/reF2JN3CqUXSnEqLwcpU39jzQ/9+zJccKvtiW6j0cdUWbAZ/VoHoFX3CTil5m4qxsQeLRHQqj82F11Rc0T1a/QxXTz0X/y6WQBCRryPun9EzI1ZI4IR0PRJrD52Sc0R1a/Rx+QuzcMfugUhoE0PZBy9rGZ/cvnISnS/vtdqHv4nHOVxHt0B3zNddzxzIfp1bIHg0L6YNCMB81MWIHHmX9E3NBgtOvZHaub/1JZEt8eYlEuFB/Dxe9Pw2uiXMWrUy/j9a3/EzOQlyC2qubciuh3GROQTwP8BukFBPG6gW8cAAAAASUVORK5CYII=)
Two thin rods of mass
and length l each are joined to form l shape as shown. The moment of inertia of rods about an axis passing through free end (O) of a rod and perpendicular to both the rods is:
![](data:image/png;base64,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)
physics-General
physics-
A right triangular plate
of mass
is free to rotate in the vertical plane about a fixed horizontal axis through A. It is supported by a string such that the side AB is horizontal. The reaction at the support A is
![](data:image/png;base64,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)
A right triangular plate
of mass
is free to rotate in the vertical plane about a fixed horizontal axis through A. It is supported by a string such that the side AB is horizontal. The reaction at the support A is
![](data:image/png;base64,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)
physics-General
physics-
A cylinder of mass m suspended by two strings wrapped around the cylinder one near each end, the free ends of the strings being attached to hooks on the ceiling, such that the length of the cylinder is horizontal, From the position of rest, the cylinder is allowed to roll down as suspension strings unwind. Then calculate the time dependence of the instantaneous power developed by gravity
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)
A cylinder of mass m suspended by two strings wrapped around the cylinder one near each end, the free ends of the strings being attached to hooks on the ceiling, such that the length of the cylinder is horizontal, From the position of rest, the cylinder is allowed to roll down as suspension strings unwind. Then calculate the time dependence of the instantaneous power developed by gravity
![](data:image/png;base64,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)
physics-General
physics-
A man pushes a cylinder of mass
with the help of a plank of mass
as shown. There is no slipping at any contact. The horizontal component of the force applied by the man is F. The acceleration of the plank
![](data:image/png;base64,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)
A man pushes a cylinder of mass
with the help of a plank of mass
as shown. There is no slipping at any contact. The horizontal component of the force applied by the man is F. The acceleration of the plank
![](data:image/png;base64,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)
physics-General
physics-
In the fig shown mass of both, the spherical body and block is m. Moment of inertia of the spherical body about centre of mass is
. The spherical body rolls on the horizontal surface. There is no slipping at any surfaces in contact. The ratio of kinetic energy of the spherical body to that of block is
![](data:image/png;base64,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)
In the fig shown mass of both, the spherical body and block is m. Moment of inertia of the spherical body about centre of mass is
. The spherical body rolls on the horizontal surface. There is no slipping at any surfaces in contact. The ratio of kinetic energy of the spherical body to that of block is
![](data:image/png;base64,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)
physics-General
physics-
A solid sphere rolls without slipping along the track shown in fig. The sphere starts from rest from a height h above the bottom of a loop radius R which is much larger than the radius of the sphere r. The minimum value of h for the sphere to complete the loop is
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)
A solid sphere rolls without slipping along the track shown in fig. The sphere starts from rest from a height h above the bottom of a loop radius R which is much larger than the radius of the sphere r. The minimum value of h for the sphere to complete the loop is
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOwAAACeCAYAAAAxOebeAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAFxEAABcRAcom8z8AACZ2SURBVHhe7Z0JvE5lHsdbMBOTpbJUU9NknbTYZcmaQtlDC6YFJREqTMuUZFC0kGRKiywtMiqSaSFUxpJIlrKGa0mKJKT+c7/nnud6zvOe8273vpdz7/P7fP4f7jnPWd7nnN95nue/niAWFhahgSWshUWIYAlrYREiWMImiS1btsgHH3wgr7/+ukyePFlee+01ee+992TDhg1uCwuL7IclbAJYs2aNDBo0SFq1aiXVqlWTs846SwoXLix/+tOfnH9LlSollStXlhYtWsjdd98tvXv3lqZNm0r//v3lp59+cs9iYZE8LGHjwIoVK+T222+X888/X0444YSk5JlnnnHPZmGRPCxho+DAgQMyevRoOfvss31JmIhcfvnlsnfvXvfMFhbJwRI2ANu2bZOuXbv6ki9Zad26taxdu9a9goVF4rCE9UFaWpo0b97cl3RKzjzzTKlVq5a0adNGunTpIs2aNZP8+fP7ttWlZs2alrQWScMS1sB3330nHTp08CUbUrVqVUfxhIZ469at7lEiffv2lQIFCvgeY0q9evVk06ZN7pEWFvHDElbD77//Ln369PEl2SmnnCIDBgxwNMU69u/f7yikzPZlypSRXr16OZpkcx/Stm1bqzm2SBiWsBrefPNNKViwYAS5zj33XJk0aZIcOXLEbZkBlEg9evSIaH/eeefJO++8I4cPH5aPPvpIKlasGNEGGTdunHsmC4v4YAnrguktU1WTVCVLlnScI0xA3qFDh8qJJ57oaQ9Zp0+f7rbKwIIFC3xNQmxbt26d28rCIjYsYV2MGjUqglD58uWTsWPHui282L17tzRu3NjTnpH4rbfeclt4wYhbqFAhT3sE5woLi3hhCZuOnTt3SpMmTTxEYuRkncm01g8HDx6Ue+65J7N96dKlHVIGgfPo7ZWUK1dOtm/f7raysIgOS9h0zJo1K2L0O/300+V///uf28If+BMPGTLEUVQx7Y2F1atXS/ny5T3XQZk1ZswYt4WFRXTkecIeOnTI0f7qJEJwcshuoIX2G2Wx5f72229uKwuLYOR5wu7atctxG9QJhAME0TepwMyZM6VEiRKe61WpUsVj07WwCEKeJyxa2nPOOcdDoGLFisVPoPRRMxHg8oiHlH49rj9nzhy3hYVFMPI8YT/77DMPeRBC5/bt2+e28MFv+2XH5hUy85VnZcKsZeK49O9Pk4XvvCRPjBwtr85ZLfsOOS190a5dO8/1/vjHP8r48ePdvRYWwcjzhMUMo5MHueGGGxwtsImftq2SuW+/LEPu6iyXVTjDaVvjrjdl03frZOzNVaSgOkeJ2jL4nW/EX78sjgeUfj1k+PDh7l4Li2DkecKyVjXJg6uhnzln/YcvygO3tpYa5xd2254kDfs+KkPu6SoNa1WTi85RXlKFpO2js+Vn9zgT//jHP9x2R2Xw4MHuXguLYOR5wuLFZJIHd0M/wv6y/2c58ttBWTn1Til9Im3zydllK8oVd4yWT77eJKvmTZT+XdpKm04PyDsrv3OPisTAgQMjrvnII4+4ey0sgpHnCYuzg0me9u3byy+//OK2iMTOZWOkwSkZbQuUay8TV+x396TjwB7Zsz9oMpyB7t27R1xzxIgR7l4Li2DkecIuWbJETj75ZA95LrrooqjZIbYveVYaF8loW7nnBNnkjQmICkbuq6++2nM9ckIRXGBhEQt5nrAbN250QuF0Ap166qmyfv16t0Ukti8dJ5cXzWhbu98U2ZSAz8M333wjlSpV8lyPIICFCxe6LSwsgpHnCfv9999HjHg4/T/77LNui0johL20zyTZmMAIS0pUPgj69chCQTCBhUUs5HnC/vrrr/Lwww97CIQ0bNjQcSX0Q7KEZTrslyeqc+fObgsLi+jI84QF8+bNc7ybdBIRyB4UfZMsYefPnx+RgYJ8xlOnTnVbWFhEhyVsOvBqMr2PEALa/aaqyRAWJVbHjh0jroFXlU0VYxEvLGFdTJgwISKJGtrje++9121xFDpha/WdHBdhn3rqqYisiieddJIMGzbMbWFhERuWsC5IpnbVVVd5CIVA4pEjR3rC33Yue16auGad6r0nxzTrvPLKKxGKJoQgAJReFhbxwhLWBVE71atXjyAVgnM+qU1/+OFHp+2OBUOl6h8y9p155YOycI+zOQJMdZ9++mk57bTTIs75hz/8wQmct7BIBJawLl588cUIUpnSvuO1Mu215+T+jpUkX+b2M6RJ9yHy+oyP5VvXOYrRmAwUN954o+d4XQia93N/tLCIBktYFxSr8iNWkBQpWlRq1Kgh7dq2knat28m1XQfIvyZ94ITJ3XprdznjjIxoHj+56aab5McfM0ZrC4tEYAnr4uWXX/YlV5BgjsFjiZIebdu2kWZNm0i58//i21YXNMVUF7CwSAaWsC5effXViBzD2SnYdQmrsx5NFlmBJawLCIuZxY9sWRHcHAkmwDkCryoLi6zAEtaFSVg0w4hOPiWQ0G+7LkWKFHEKZz355JNOojcLi+yAJawLk7AXXHCBQzbyE+tEZNrM+vXCCy+UBg0ayN/+9jcnGXiFChWc7IeUnezXr58TGG8VSxbZjVxFWBJ7Y/eEaF988YW7NT6YhGV0BG+88UZEVkXksssukxkzZjjheVzryy+/dK5vYZFK5BrCkhBc99W95JJL5Ouvv3b3xoZJWI5X+PDDD30r0DEKU53OwiKnkGsIS30aMjfohGJ0jBd+hNXD65YtWya1a9f2nB+hEvtLL73ktrKwSC1yDWGpiE6dGp1Mjz32mLs3NmIRFmzYsEFatGjhuQaCnzBO/FYLbJFq5BrCkiZUJxxy3XXXRc3NpCMewgI0vrfccktEHig0x5SOjJqA3MIii8g1hPWLtCFXE6NiPIiXsIDInvvvv9+3WjsfCVs+0iJVyBWEJUTNTKSGYIIhm0Q8SISwAAf/0aNH+0biUFxrzZo1bksLi+xDriDsf//73wh7qRICx6MRTyFRwipQOeDPf/6z55oIZqFY9WUtLBJFriAsRZXNNaUSpqgHDhxwWwYjWcICzD6YePTrIrGqsltYJIpcQVi/fExKyPkbj2tgVggLPv/8c6lbt67n2kjx4sVtZTqLbEPoCbtz505n+mkSRQl5lOKZmmaVsIDk41Ru16+PYB+mdg7OHRYWWUHoCfv+++87zgsmSZSgeEI5FAvZQVjAaH7bbbdFBAhw7jvuuMP6F1tkCaEnLEWkYkXPsI6NhewiLGDN/M9//jPCkQO59tprnSrsFhbJIPSEJd2KSQpTiKj5+eegaq0ZyE7CKhCIYCYoRxo3biyrV692W1lYxI9QExYHBaJmTEKYgq108eLF7lH+SAVhQTSzzyeffOK2srCID6EmLJEy0davSsgtHGsdmyrCAvyc/aJ9MPv85z//cVtZWMRGqAk7duzYuPMwxSo4lUrCgqVLl0r9+vU994SULFlSxo0b50lUbmERhNASlhf8zjvv9Lz8uvMEiiidgGSDiFZVPdWEBfg1t23bNvMaSvBJpoJetPuzsAChJSyaVpQ36qVnpCWbvvq7UKFCnr/JGhHNHpsThAWkOO3Ro4evZ1avXr3khx9+cFtaWEQitISlYjleRPoLD0nV/9HO6tE0JFRjCh2EnCIsYCTF7KN/UJRggtq6davb0sLCi9ASdsqUKZ4XvVSpUh4TSokSJRyFlE7C7t27u0dHIicJCzg3Zp+iRYtmXlMJ0T5fffWV29LC4ihCSVgyOwwcODDzBYdoLVu29GiMKZXBi6+HvzVq1CjQ0yinCavAh8cvyRtmn48//thtZWGRgVASFve/Jk2aZL7cEO1f//qX4+ivtpGKlEz75cuXz9yGGSVoHXusCAtwr7z44oszr62E3/Pmm2+6rSwsQkpYpot6sSnsrMTE4tGktrGeJTdwnTp1MrcRCPDCCy+4Z/HiWBIWkOSNPMfq+kr4nc8++6wcORJH1WiLXI9QEvbtt9/2vNSUwoDEOmFRMs2fP1/+/ve/e9rec8897lm8ONaEBeQ4vuaaazz3i6CcsmYfCxA6wjLS3HfffZ4X+uabb3Y0q2XLls3cBmFJD0MZSd2EQtZDP9PJ8UBYsGfPHsfso+5DF5K8sd8i7yJ0hCUroW5/RTDX4Nyvr2EZlVDaEFjOelZtZx27ZMkS92xHcbwQFhDt88ADDzhTfXU/Soj2+fbbb92WFnkNoSPs5s2bPZpfPJpUorXzzjsvczuEnTt3rhM0/pe/HK3bSnvWtiaOJ8ICZhKjRo3yjfZB4bZixQq3pUVeQugIS44knVgUocLljxfcJOycOXOcY66++urM7chDDz0UQcbjjbAK3Jf+wVFSuXJlWyYkDyJ0hB0+fLjnxaWeDlNIFDImYdULjclHP6ZTp04R8bHHK2EBSd64H/03IPxea/bJWwgdYYkv1SN0MHkA1rYmYQlrA4sWLfKsY7HPHj582NmncDwTFlAhz8/sw/KAPrDRPnkDOUfY/btlx84fxEuTxMHIiB9u8+bNHcXM7t27ne14MJmExSEBMF3+97//7VQH6Natm6xbt87ZruN4Jyxg/R5k9qFUiTX75H6knLAHtq6Qdyc8Krdf10pufmCSbHK3ZxUmmaIRNh6EgbCAKge33npr5n3qQrihNfvkbqSIsEckbfm7MrJfZ2lcvYKcli/jhSrTaqikyqU9rxAWMMtgdoGtWd2vEsw+mzZl12fR4nhDigh7UDYs/q9MfOYxua1JWcnvvEwny8UdR0iqUo/lJcICAiBIe+NXogQ7daIV6C3CgRQR9vf0dWOGEuSH2fdKhcIZL5IlbPaDotX671ZCMAF2aIvchZSvYQ8te0pql8jQ6lrCpgYQk/tV967k3HPPtWafXIaUE/aXFWOkfskMIljCpg54PvklecNTCg15mH6LRTBSTtiDK5+RBqUynO8tYVMLzD5+hcGoQEBtn3iq+Fkc37CEdZEbCAtI8uZn9uG3UdtH2a0twokcJ2xmXfJ9m2XetPEyYshgGTZ6osxfm7UXyRL2KPbv3+84l/iZfdq3b+/E3VqEEzlK2EuuHyUb5Yhs/WCMdKpbTkoUOUVO5EXKf6qUqtBMhry1KmlPKEtYL3C9JBbYz+yDi2NeNvvgxkkU18GDBx3vMP7l7zA87xwk7ElS9bqH5fUZI6VDzQulSoMW0vnGLnJVzTJSyH2RClbvKe+tO+gemRgsYf0xbdo032gfsnTklSRv2Kypw0RWEsxgd999tzPTwFX1yiuvdKK5OnTo4CRGmD59uqxdu9apO3w8+mfnEGHzp78kBaTEXyrJVd37yDMzV8svLhd++365jOxYXvI5L1Jp6fPa8qRGWUvYYEBMvyRvEDk3m31QspEmiLRAlSpV8i3/6SfkA6tRo4azrCBwBF/04wU5SNgTpGilTjJ51T53j8IRWTn1TjnP6az80vSR2bIrCZ5YwkYHZp969epl/j4lTJlzW20fprnETd9yyy1O9XvzNyci9E+fPn2cGOp77733mFcczNE1bOVOo2SDu13Hho+GyaUFMjqoZr9psjWJIdYSNjbwMfYz+1AhYdCgQbnC7EPdXRLv8fzN35kdUq1atWNaTiVHCRtk1tk870mpf2pGh1S+43XZnESUmCVsfMDsc/vtt3t+K0LqHJK/hdnsw3rdb+qfnUI/4YhyrHBcEHbL/NHSoEhGh1zSY4psTuJDbwkbP4j2IW2qn9mH6nphi/ZBw8t6U8/1ZQrPlsLabdq0cZxImDJ/9tlnTmJ5/oXsREARZ00lBvODpgu5op944gn36jkLS1gXeYmwQAX1+73kuDiGJckb5qsBAwYEEozpfs2aNWX8+PGSlpYmP/30U+BzpU8w8ZCU/uyzz/Y9nxIyWo4YMSLH3xFLWBd5jbAKVID3q+0TBrMP5hpGy6D1Kr9hwoQJEfm7ogGnEsITo42wStA6qxRFOQVLWBd5lbAAYvJyq9+uhP48Xs0+jIaQ5dRTT424b9aZ119/vWNPTQRff/211K5dO+J8bGOqbG5HmKH4pc1NFSxhXeRlwgKmwEG1ffCYOt7MPuSi1qsVKmEKjAkmUY032mU/sl566aXyzTffOKN0v379IvYjvHdr1mQ63aYUOUTYDCJc3HHkUV9iDVsWPC0NFWFvTyes1RIfE6BswgNI9YESHAlQ6hwvSd7Ia0XJFfM+KXbGfSbq6IAHlB9Za9Wq5Yy6Cvz+nj17RrRDMCXlxEct5YQ9tHC4VDs940dVaDNUVrrbdWyd/7jUKZTRpkzn52RjEt6JlrDZA8w61PBR/aCEaSZRQJDlWIOpsF4vSQn3nch6FTAy6hUOlbDNb0q9d+9e6dKlS0R7RvapU6e6rVKHlBH20P49sm3TCnmxZ203p1O6nFlfHp6+XDZv/U72Hzoiv+zdJVs3LZXn+9SVk1WbUs1k1OwvJW1HYsZpS9jsA5rSIUOG+Nb2ad269TGt7bN+/XrPc1aCFxfvQCKA3F27do04F6NttCkuU2QcKMzjqlev7tQuTiVSRNjfZd1HL8id7RpKzaqVHT9O5JJKVeTSelfINT1GyvxN38vyaUOle8emUrda5fR9R9vUbNhK7npwUjph3NPFAUvY7Mdzzz3na/ZhrXuszD5mFQeEKTvVERIFDv56iVIEsrKejYVJkyZF2LF551jvpxIpnRL7v/C/a9v5v/tfA4lyxRI2NcDsQ24o1S96/+R0kjeqO1BTyLwXom8w8SQKFFNUgWDtyxQbc86qVavcvdGBPbdz584R99KyZUtnhpIqpHwNm1OwhE0d0MjSH6pvlPz1r3/NUZPGjBkznOeq30OJEiVk5Uo/zUh8wEvq5ZdfdhwrEl2fz549OyLeuFSpUplF2FIBS1gXlrDR8eWXX0qjRo0y+0cJZh/yI+dEX1133XUR10cRlpj2+oj8uH2jrPnqK1m7boNs2bLFcebfs+d7+Tb9/5u+3Sbf/Rjf+cjs0bRp04h7YtROFUJLWKYz+ksSD2E5Jkj1bgkbGyibqCyg+kgJa8j7778/pVNBtNflypXzXBelGD7AieGwbF76kUwZM0hual77qH6lcjWpU6+RNK5fRy6tWUcaNG0nPQePl082/uQe54+RI0dGjPqQOFVBFKEjLF9DNHsoC26++WbZunWrsz0aYZn24MLGMXTm8uXLne06LGHjA/1PDR+9giDCGpDRLlWhZwSilyxZ0nNNtLKJ56f6XQ4f2C97dm6W2YObZVgn8pWVriNmyKLFS2TJogXy5ui7pd6f86Vf4xQ5s2YXeWnhjvSj/IGCiqWBfl/ly5eXZcuWuS2yF6Ej7IsvvujpHL5wIFq5SdZg+leQFw4S67CEjR+MpEOHDnVsj6q/lBDtQ7rV7AaBCub1cFYwn2Mi2DXnASnHuUo2klHzdfvt77Jl3hhpWirjOue0fEQW7fB3xuAdIbhAvy/uc+bMmW6L7EXoCAtB9c7hBWEtwTrGJKwq6Pzggw96jsHwbbquWcImjueff943yVvDhg2z3exD1gfzOng1ZQVpHz8kFTlXifoy4v3v3K0uft0iE3tUEidJYME68vjczYGjLH7L5r2lKiggdIRlmqt7uZQuXdpxH8MdzSSs0tY1adIkczuCLc+EJWxyIGmZ3u9KWBcmYxsNAuYS/fyYYl544QV3b3JIm/ugXMD5/Agr38uHw69yc41VlPtmrpIgx0PeJ9PzauDAgSlxVQwdYbdt2+ZZy0AyXgzIZRKWKBS8WfTYRhQV2BZNWMImD5YcfvZRngf9mh24/PLLPedGO/3uu++6e5NDVML+miZT7nBNWcWbyviFwR5MEydOjJius55PhRIudITFYE16Sr1ziP5n+/nnn5+5TRGWbAJ6CFaFChWs0ikFwOxzxRVXZPafEjylRo0a5bZKDgSpM83Wz8tHOKsj+FHCNpDHP/Iqy9LmPiENinOtAlK1x/Oyfn/wu4Cm2gzzIwFcon7N8SB0hGWaMXjwYE/n3HDDDY7JQVf74zaGZvHxxx/3EJE1LwoqE5awWQc2zU6dOmX2oRICvTH7JKsg4jgz9A/CZtVBIZOwxevII9PXyw8/7pXd366Q6c/0lyYVzkifDheTOp2HymfboofqMWMzszPedNNNjm4luxE6woL33nvP0zkQlS+87hcKYZmqmcZ2UlX6wRI2e0A0S9++fZ3oHr3f6dtkzT48B9NpA48iUrlkBRD2wnzp5ytwhlx02VXS6spaUrpUMSmYvmw6r1EvmThnpew+EHsdyruDLVq/v+7du6ckHDGUhCWSggemOoeXA7c1prtqGx2IgzYByGobJH7llVfcs3hhCZt9wK8Xs49fTuBrrrkmqdo+pkcRz/e1115z9yaHoyNsbRn0nw2yb8sHcnejjKD4s+r2lXl73IYxgKeXGdmEVjsZ/+ZYCCVh8SKhvILqHIiGil83YBcpUsTJEFCmTJnMbRi0P//8c/csXljCZj+I9ilevHhmnyphPZqoYwFOMuZ5Hn30UXdvcvCsYedkjPw/Lh4nzc4l8X1hqXvnFNkexyye9LDmvZGgLRUIJWEx4ZAGRHUORKNGip4yBC0iMZIQV21DKRKkCLCETQ1Y32F6U/2qhBxSifh6P/bYYxHTbPIrZ8V0cpSwupb4sHw+vquckz99e+Fy0mviSok1sTXX19xnqoLZQ0lYQIfonYQ2smjRopl/E8XB1113oevVq5d7dCQsYVMHdAm4Eaq+VUK2xsmTJ7utooM8wqaTRrNmzbLks+tP2HQc3iTju1dxEi/8qVwHmbIyOIqHgHV9KYbwuz799FO3RfYitIRdunRpRO5YfeFfrFgxT/Ej7GTRDO2WsKkFcaYQTPWvEj6y8Zh91q1bF+GgAYGxBCSL7R8/lEHYkvVl5AdeO+zhjbOkR81i6dc5Wcp1eFS+COAs740+i0Muu+yylLhngtASli+b/gIwkur+whBU/5tKbdHWTZawqQdOL35mH54TWffJABFk+mHqS4Jz81iCOvyf02HZtfZTeXvaWzL/q+2+FRF3vf8PKcN5itWSYbNNTyeR9bMekTqncZ2i0vT+N2STMTfmun5B7N26dUuJlxMILWEBmQb0jtIJh6uY/jcO2hjgg2AJmzPAwQUNKq6Fqq+V4MGG6SdomvvUU095nhFStWpV50PgxW+y4+PR0vKColIgfwEpXOYKefT9ra5r4W+yd+dmWbP8UxnXrZJ7noLSqPe/ZcmK1bLl+wNy1M1/l8wa0kYKO23yS43Og2TqnEWyasMO51wrli/3KDURBoopU6ZkHJ4ChJqwOJ+bPpxBQkheNFjC5hxQGpLkLagU5LBhw9yWXjDNPOussyLakzHCiz0yqW9DKZjZpoBU6/ay7HX2HZKv502Tpwf3lS7tW0ur1q2ldatW0ub6bjJw8FPyzudp4nEo3LVUnul/vbRscbXjYde2U295fNI82YcDz0PeoBKEnFDMFFKFUBOW9YtfviFTsL/GqjhmCZvz4Jn4BQ6wjSJVJphm+pl3cJjxFvDaLx+P6CjFM9sUlqsHfyBJ5KcPxP8WLoyobM/7k8psEyDUhKV0ol/aElMw8cQK97KEPTbwC01DKlas6ORMMoE7oq6bUEKMs47D25fJ+Af+Lldd0VRuGDBWFm1LPm7WBK6tHTp0iLgHtMN43KUSoSYswAXM7DhTqBkaK3LCEjbnQfI0v0JcSrCr462mAz0E61yzLdNrnqEHR36WnWnbZV82OxyxljY9m3h3iLtONUJPWEwCfgoMXXDEjgVL2JwFfYs2VX9OpmMEQhQMz1h/FpTWMPM7IZA/qyF3sYBrq18BLpSae/bE6cuYBYSesBjlKdRrdqASzD3xVMy2hM1Z8Nx0RxeEaa3fGhUi33fffR4vtaByHTgxqEwj2Q2cdVhemddkip6qlDAmQk9Yvmo1atSI6EQlKJy++OILt3UwLGFzDkxrCXPUnxNrVjKHoFgKMvuw/FGlMIj6YeZktkHwKacubDQzXiIgAokwTb90OAwI/fv3z7ZrxULoCQuCFBcIGsR43NcsYXMO1JwtXLhwZl9DTt2Ug9mHtCt6GyWtWrVyatsAYqDNZAZKWGMyYidaI9YEHnXR3i9mBBA6p5ArCEtitqB17I033hhXXKIlbM4Azb6ZmYL1X1pamtviKMjG72d3pbLc4sWLnTa4LOIKaLZRgg8zCQ/iLcGhQKYS8jIR4eV3XqRdu3YpL35lIlcQlvWQXxgXwlonHuJZwuYMKO2hO0ygwBk3bpy7NxLEOfuRhm0kMgBMpf3cFnW58MILnSk0Gl5SB+mzLp7z9u3bHTPS8OHDnRFVTzfkJ0zpd+zY4Z4h55ArCIv/6QUXXBDRqYy6QfGvJpim6YQl659F9oO80vpsCNe+WAHtTEvr1q0bkbycJAYq2gcCEv+sB3z4CVNlzEWsmatUqeKUjeRflk5EeMXynCMqDP9lXCyPBUJJWKa4GK9ZOzDF4mGZaTARvsI8bBQUtA8SFFd85XXC8kXm/DwY2lAoCZezWOeiPVUImCpxb/o5/ITzcV7O77ffTzhftONUv3AP0a7PPv0+zf0cF++5lKj7ok/NfRzLKKZ7p0EASBd0XvIiYUPnw8vU1zT9EKHFqMnx9MWTTz7pENLPRJQV4SNTtmxZ5155vgwSfvcbS7KamC1UhGXqwhQJv+A2bdo4WSf4QiJ0pvkFRqvHyMuaifUG0xhd2EaBYqZT2PD041F4qDYIX3i+ykHnUsK+5s2bO2k/WT+hJAlqz3bOx3k5f7Tz6kI7ci3z2wjSV9vUv3y80JwzS2jRooXvednGPtrgQO93n6qNGoWCzqWEfZRs5L7oU7/zcR1GMv05Mf3Ec0hvqwvHUdOHJAWmKQjhufEbOnbs6Ah96dcuK8I989u4j/bt2/veZyzhXaNaAcEByaZADRVhcS/0U0JYsRImwW7r53YZD0JFWAzXQdpgK1bCJMlWag8VYdHKMS3y6wArVsIi+LYnW4Q6dEonfqgZksX6gix1ZGAnOZvp6wnJX3rpJcf7hX9xKCextdmOdRSxlaSSoS1rJ30/grcN/qS041zqfCg7TLc1PG6oSoCiQrXFrNGzZ8+I0g5EHXFedY+6cC3WPeT7xXNLP461LMcQG2wGU7MOxwFBv/4bb7wR4WzAWp/+4Br8lrFjx0aEjvHbMHno5+K+MIfhE2w6w7OO1n8P/3Juv+gq1obso+SFX8gdge3Ez6LJR0NrXovfyTaSoaGF5jeSv8vMFUzCefqfe+FapFzV9yPoR9inPwf+fvrppyOWYzxfMjea/YvDhqmtZl1Nf/COYjtOFqEjLKkzzc7Q013iKK6r5nloGMFNYALQ26G00k1AGzZsiLD/8UIEBSeTh1efrqNx9vNhJlctHwL92jz4BQsWuC2CgaJN12SjKPvkk0+cfZDNDAgn7YoJ7h/HA70dHyq8ixT4YJkfFL8CYoAqgHw09PtC6eRnTiPG1SQjaX7Q7irgLKF/SFEoPfzww5k2cT7I+rUgEWVF169f77GtknpU72O0/rr5iLrCZG5U+xEUeFQv8AMDgh7Wx33xkTaBiyJpY/R7RGutnlNWESrCUjzX/PLzsigvGTIS6Nn/EWqcmH6e5HbSXxw6n3QzCpDKdELnBQ5Ky7lo0SJPYnMEUmLiMMEHRX+YXJtRIxbGjBnjGV15GXmRAS+Z+btr167tmwgMrzCuqdoxuuq/i6+/OVIzKqrC2Sa4d/33YE7xS6qGScicsdBnymMJ4HJo/g60zera3Js5yg0YMMDZp4OYWb2dIr0C5DdD9Hi+b731ltvCC+7RTNXKLAXzjglGeL0/kOwMuwsNYSERboZ6R5CtbtasWc5+nMbN2rF8qc10k7Tr3bu3px0PQw9wf/vttyOCpJmO+qnisauhstfbMkX3y+bHNcyUmHzV/YitA39YTD/mcSqXEVNsfR8jLVMzE8wazOLD3LsCoyxTQn0/0036ww98qMwRE5MW5NRBnzNq6+0QXAYVsK2bJGJ2pFcavOuuuzz7ISV2dh2cx4yRZqakj5yc0zT7kOPYLwEctlNMRXpblgd+EUFUpDCfE6a9eHzZ40VoCEsMpDm6slZQoycGdr0sB3LbbbdFFG7mwek5cvn6QkYdJgGZTjHl8gNubnoCc0Y+1oN+oKSDfl5eyHjU+0zH9OP4ECkSMc1jOq/vZ1Zh/m4AifXRFWLPnTvX3ZsxgzFHEl7koEgUPZk7wmiNm6gJnChMZWGtWrU8U2F+jx4Nw32yNlZTYUZX012QZY1ZDoMPpWnnZf2rgBOImfibUR2y+YG1tb7UYPTUZ2MKkJ2oHf28zIjIp5ydCA1hGR30Uhx8ydimwJdVT3uKosJv3cBLojuL8xKYER1UaFf7WS+jTAiCmR+ZFxG/VD+wplXtEDOtSRCYDutEY32mwLX0aSTT2SBHd4K79Wk1ShhFCEBeJJ38zAZUZIwfUEKptkhQPiNGXP0jyIdKzYwUIJ9+Lu5D/0iiu9DvnRhov7xPKIf08/AB0qfzON7gXaX283yDTCx8DMxRH0cTv+eLbsBcSnCs+UHJKkK1huVrByGYdplTTl48lEusLZj2oZELGhl4WViXsT7yW7fwwqOIwVuIOMho7mQ8EDS0XBMvHnOKpoOpEaM5an3Iytc+HnAcX2+8eZj+moovtKdoIVnPo4AJAtM7wtg4Dw7u+gdPAa0v56L4lD76+oEXl48HXlDcn9+aToHpM5pjvKZQLOlKLqDPItD40qc6mILqWniux1TbBLZ6fQ2Jll4HyjCdWCjygp4v5x80aFBmW9a5QSU4WNboUUjMypixZDdCRVjASxJtTcAI6heqZQLf2CCNL+Dl4zx+L4UfaBtPihDWwUzLE3VNYwbBmtVvqguwUcfzAeADw3nMdaYOzhXvuov7YQQLSgCug/6JzCGcAfobxRDuj8xEzJGJvzEj8UFG+x207ud+8C3GbRXrgVmjlQ8F028UYJh+grTCCrwnZLvguphv9BmJDrZTKJyIIJKlZ5dW2EToCGuRe8FLH/RBUmB/PB/RWOfhAxjvdJXrxTqfAh+ueD5eycIS1sIiRLCEtbAIESxhLSxCBEtYC4sQwRLWwiJEsIS1sAgRLGEtLEIES1gLixDBEtbCIkSwhLWwCBEsYS0sQgRLWAuLEMES1sIiRLCEtbAIESxhLSxCBEtYC4sQwRLWwiJEsIS1sAgNRP4PjL8j58moSjMAAAAASUVORK5CYII=)
physics-General
physics-
The acceleration a of the plank p required to keep the centre C of a cylinder in a fixed position during the motion is no slipping takes place between cylinder and plank)
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)
The acceleration a of the plank p required to keep the centre C of a cylinder in a fixed position during the motion is no slipping takes place between cylinder and plank)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQsAAAB5CAYAAAAuyAQsAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAFxEAABcRAcom8z8AACE5SURBVHhe7Z0JuI7VFse7t+letzmalUqlklIoMmUoQ4Zo0CRlzEURlfHmXipRiMyJUoZCZoVQkQ6RknkmleGYSjixrt/27td7Pt85Z3/f+c68/s+zH8c7f++793+vtfYaThGFQqFwgJKFQqFwgpKFQqFwgpKFQqFwgpKFQqFwgpKFIkfjt99+k19//VUOHz7sbVEkBSULRY4E5DB58mR56KGHpFatWtKtWzfZunWrt1cRDkoWihyH7du3S48ePSR37txyyimn+O2tt96ShIQE7yhFKJQsFDkKK1eulLp168rf/vY3QxDnnnuunHnmmebvdu3ayYEDB7wjFaFQslDkCBw5ckSmT58uxYsX9yWJm2++WV555RW5+uqrzf87dOigZJEMlCwU2R47d+6UN998Uy699FKfKCpXrixxcXFG0rj++uuVLBygZKHI1li+fLk8/fTTcuqppxpCuOCCC6Rly5aybds2s/+HH36Qa6+9VsnCAUoWimyLzz77TEqVKuVLE6gdQ4YMkb/++ss7QmTp0qVKFo5QslBkO+zdu1d69eolV111lU8U9957r3z11VfeESegZOEOJQtF+mLvOon7YoqMn/y1rNudIHJkj2xYtlp+3ndQjniHpAarVq2Shg0b+iscZ599tjz33HOyceNG74jEULJwh5KFIt1wdNt8Gdz5Zenca7iMmzBShoz+WpZP6y4NGnWRSat2pZosZsyYIWXLlvWlCQyXAwcOlD///NM74mQoWbhDyUKRTtgv3/SoKcUqtJQxK3B82iuLP+gjL9e4VvJW7yZfbz54/LAoABm8/fbb/qCnVaxYUebMmeMdkTSULNyhZKFIJyyRbtUKSImnh8nS/d6mrSOl7q1XSblOs2TrIW9bhFi/fr00adJEzjrrLDPg//GPf0jz5s1lzZo13hHJQ8nCHUoWinTCPOl8301SttFIWWbH4/qBUvv2qtJp6lqJZoh+8cUXcs899/jSBM5V/fv3l3379nlHpAwlC3coWSjSCdtlVJPiUvLJPvJt/PEty3vXloJXlpPOxzYc2h0vOw8eEpfIDOI33nnnHd+ZioatAvKIFEoW7lCyUKQbdsW9Iw1rPSKt+06Qz8e/LU0eflratWsq9Zo2l/rPD5K4335P0cj5888/S9OmTeW8884zA/y0004zagirINFAycIdShaKmOH333+XDRs2yOzZs6Vv377yzTffeHssju1fPFumTJggEz4dJ1PmrZH4vVtk3tj35L1Ji+W3P084S4UDfhLlypXzpYkrrrhC+vXrJ7t37/aOiBxKFu5QslBEhPj4eOnSpYuZzf/73/8aH4ZHHnlEKlSoICVKlJDChQvLNddcY6I5b7/9dvn666+9MwNIOCD7D7gvlBIExhJo/vz5faK4++67jdpx9OhR76jooGThDiULhTP++OMPef31132HJ1Ye7OAN1/7+97/LyJEjvbOjA5msmjVrJueff75/zQYNGsi6deu8I1IHJQt3KFkonHHo0CET0h0khOQa4eAMxmixcOFCI7FAEFzvoosuMv4USDexgpKFO5QsFBGBLFMM2FtvvVXOOOOMROQQ2p555hljx4gUqBbDhg2T6667zr9WkSJFZObMmTHPZKVk4Q4lC0XEYDCTr5IBjZHRDujQ1qlTJ+8Md+zatUtatWqVSO144oknnJ2sIoWShTuULBRRYerUqXLHHXeYpUsGmv3XNtLWDRo0yDvaDd9//71UqlQpUe4JktakZrUjJShZuEPJQhERGLishrDaYUnh8ccflw8//FCKFi3qk8Vll11mCMUF5JfAEHrDDTf45xcqVEimTZuW5in6lSzcoWQRIei8dKj9+/fLnj17jLGNxt9sY192rUGxYsUKkzr/9NNPN4MLwiDJLfkjABmzrTGyWLFiRlJICZDPyy+/7KsdSBUPP/ywSXeXHlCycIeShQMgAZyNPvroI2nTpo3Url3bzHx58uQxnoQ0/sbo9+CDD5pjRo8ebXIocG5WB0ZFamzccsstZlDRbrzxRvMbWSGxgDDtMifvICXS/PHHH6VatWo+wfAeX331VXOd9IKShTuULJIBa/yjRo0y5AAZ2E7t0jj24osvNueOHTtWduzY4V01awGpiQI8Vu1g5r///vtN7spw4Phly5YZQ2VSOHjwoHzyySeGcOz7IuXdpEmTEqW8Sw8oWbhDySIM6PDMmiVLlvTrS9BwRmKtn+jG0qVLG12drEyNGjUyf3N8vnz5DLFYxyUaxFGmTBlDGuk5a6YWqAKPPfaY/w6Y+ZEYUvMbeLesktjYDpZfH3jgAZNYNyOgZOEOJYsQ/PTTT0ZntgOddvnll5vU8X369DEzKqoFMyBuyCwj0vibbexDVycHJHkfMfTZ6zDoWAZE909P2Gd1BeoDyW5vu+02/9kxPn7wwQepcq+GEJC07DVZ7cDJKyMJVMnCHUoWAWB9xxZhO3PevHlNDAQdKlosWbLEuCcHSQMHI1LApSWYwQnkGj58uAnnZhlz1qxZJlmMNUiGA+dBdAxknhW147777pPvvvvOOyJyoHZ8+umnxqZj3wEqCKpIasgnFlCycIeShQeIwhahwWegSpUqsmDBAm9v6kFAFYPO+hBARJ9//rm3N3ZAKvjyyy/lqaeekn/+85/+4LSNHBDPPvusGagQQxCrV682NTbssagKBIpRpCdaYKvp2rWrTz6oHRg1k7J5pDeULNyhZHEM3377rR/RiK3h+eefTxNHIAZOixYtfDfpggULpmrGDoUt+ItdhevTMEziZYmxNdRxCnsLAVmsaCB13Hnnnf4+1I7Bgwd7V44OqFvYPOw1seW0bds2WeNnekPJwh05nizQl21qNgYTBrxI9PtIgf0AMrK+ClWrVj1pho8GzNQYWe3AZPmyVq1aRv1A5Rk/frx07NjRBGYh1djj8Jgk1By7DP/HrsIxJ+eicAfSDSsbQSetAgUKyIgRI7wjMg+ULNyR48liwIAB/kzPLEgYdlqDHJEMZLvKgPditARF5x4zZsxJxkhIItwyJDYCjJfly5f3VSLbkEBQO3799Vfv6MiB9MRSK1IE14SAIUQiSDMjlCzckaPJAnHYGt2YbU9avkvYLRtXLpdVazfIlh375MCfsfPMXLRokV8xC2/HaOwCmzdvNh6UFNLhOkgr1atXd5IKcBgLzvw0bCqpWZlA7ahXr55/vdy5c0vr1q2NepRZoWThjhxNFngl2hTyL774YmB2PyqHdq2UGWNGyOjxk2T6jIkycsxHMnr6am9/bIB/hp3d58+f7211AynmatSo4Q9MjLN09kicv5BoIAzrE5IrVy7zHshzGQmQVqZMmWKyV9nnYbWDqNSMXu1ICUoW7sjRZIF9gsGKGoKR0yLh940y462G8mT7T2Wz4Y89snTOKBk8MsQY+ddeWf/dXJk5J07W7Ei66lVSINDKrhKQgcolVwPLnqS7D+Z6QDL5+OOPvSMiA7YO1IZgyjpc1l3iOgDSWffu3RMtDaN2zJs3zzsic0PJwh05liyQIjDk0UnwyDxhoT8iu1aOkefKlpcucScGbwIOV7sDhsi9W2TOwGZSuXA+uTRvQalYr7tMWxWZlR/xnHvzDHXq1EkxjgSPSvw+rHGUpVESzBBjkVqwdByUDFgZmT59urc3PFA7WGq1theI74UXXjC5LrIKlCzckaPJAucoOgmu2CcK0yTItiV95dHCNaX/6qRE6GPHzOklNS4+PrCOt/OkQttxsjnCRE7ERHA+3p7J2QsmTpyYqKAOSXFxnoomE1VSQMpAqrD3wHX93XffDWsoRSrivdljWe0YOnRo2GMzM5Qs3JGjyYLkLXQSYjpOkMVR2b9+qrSvdLc0GRcwzB05KHt32sF8WJaOaS2FvYFi281P9pYFEdpAb7rpJnNuUmSBxEMkpl3atMeyopEWYCWEpV2bjPecc84x+StsdCnSDwlpIBL7PCy/utQVzYxQsnBHjiULDG94adJJWAkhwtTHwXhZMrq91G/USQaMmiifffGlfPXtYlmx6YSjVvzSj6TJbbn8AXPKadfK472/lEjs/ojrV155pTmfmJFQKQGHrUcffdS/B6sepJyLVWbrpIB7NoRwySWXmPui9pCKn9R2GGWtGvSvf/1LWrZsacL3syqULNyRow2cRD/iB4CRc+7cud5WD3/tlB+mjZIPx02TL778RpYs3yC7glJDQrysmP6OPPdEDalUrY407T5Wvv8lsuq+uFzb0G+S4FoRHiJ7//33fTWJhrrC6kIwf0Rag8AxPD+5PwbMoBrEAMOXgwrmWRlKFu7I0WSBi7NdOm3cuHFUjlGHd2+V9Zt3yEHv/5EAj0vC12kEnAGWLfFNuPDCC/2BSRRspEursQIShjVg2oZhmKpj2QFKFu7I0WSBnYLaFnQUHIji4uK8PWkPqmnZGA50fpZNCQCzqhENwsBesG3bNu+s9AcrLZYssGNgzyDgLLtAycIdOZosAJmwrLs3OSvSI6MVnpdU/WYQovcTR0EYeTBzFEuXhHVHI+3ECqhFLIXyPNgpCDzLDmkCg1CycEeOJws6B3EadpDWr18/psuRoSBoDGOmna1xz65bt66fOYrtqEQZHcKNNNO8eXNfTcNmEqlnZ1aAkoU7cjxZAJyLbHg29gNsCWlR1AYCIDs29+BeOGTZpVMay6NUBY9FFGpqgIpkHdZoBIWltmZpZoWShTuULDwQfBVUA+666y4TzRmL2AZCtslYFVzdQP2wrt40fCcy2mjIc2LQtIOHxnvAkzPWZQMzC5Qs3KFkEQAGTuwWdqCwbFizZk2TCyKa0HX0exL/kg0bA6q9Lsu11lcBewl1MzLaVwHfDdQhq3bQcC1ftWqVd0T2hJKFO5QsQrBp0yYTYEYEph00kAbBWmS5wjeCDhYujyUemCyBQhCkriOi0+Z1oLGaEMxWRTAYtUhOtpEclQM75sugF5+VhvXrm/gPWv36DaRBoybSrFUHeXP4dPnxt9iEzEOGwXB11CGcsNKybGBmgZKFO5QswgCJAPdlDJFWArANCQFXZwx+iOg4KtGweWB/IEdF0EeChss059lVFxq2i6RT6h2R3WvelVqXnCUXFKgszdp3lI7HOnKHDvzbVlo8VU2KXn2p3FCllXzwffTGWH4njmnWU5NWsWJFky80I1dh0hNKFu5QskgGxGXg2UmCmdtvv90fUC4NkilRooSJJoVArEQBcbz22mspJIQ5RharB0rVC8+RfPf3kB/37TcD+3jbJ/Hb18lnPevI5aecI2VeGCubvbMiAYOEfBiWwMhpgTq0ZcsW74icASULdyhZOIAOxLIhORpwcWZQIRmUKlXKiO80ig7hadm+fXtjzGR25m/iTuzqB4FrJNxJ2WUbyWKQVLvofLnuwYESbsFy88zOUuSUU6VQvcESiVUBgy3PFzTmkvEb1Sk9UgpmNihZuEPJIkIgntOh0Odx4CIAjcbfeIQShIWxEJuFXe2ALKgd4u75eJwsql98rlx13ysye/U6Wbd2rVnOXbPie/lqbG9pVvEqyXVxCXluxEpxtVzwnNhdkG54Lnw6CEnPaJ+OjISShTuULGIMQsexZdD5aKS7wzszMkevI7Jn3VB56JIz5PSz8shV+fObTFamXZtfri9wi9xVqZ50GbVQth9KeWkXgiMNH7YV6wxGABvZuVJTEyQ7QMnCHUoWMQKrI8RxWGmCQcngJN4jcmMhksVgqXHxeccki//IrDVrZe0xqWL16jWydv0m2brtN9m55w9JcHABofP37t07Ufp/7C9kxsquvhORQMnCHUoWqQQ2APJVIs5b2wQOV8RURB8AFrBZ1B4QlQETrD2mupD2zhpXScOHOpTW+TCyEpQs3KFkkQrQsT788EOTUo7ORrPFdFKXXu6IxK/qJ/eedaZcXqWXrI3QiRS7CcFpwRUc1CFcybN6/olYQ8nCHUoWUYIsV6TNt2n0+ZeAtNgYC4/I3o2jpfFdRaRC0xGyOQKyYLm3c+fOpiIZz0Vin3LlypnVGcXJULJwh5JFhEBiwFiI8xIdjIaX5v/+9z/jBxE7HDUqTiSxKTh51a5d238uCAN1KFHKQEUiKFm4Q8kiArBciigfTJ6L5+aECRO8IzIGkBRp+PCXsM9FCUPS4imSh5KFO5QsHEERIgKr7GAkeS7Gw4w2FhLLgvRg1SHiT1CHXIsE5XQoWbhDySIFYBAcPHiwbwOgkYeCmh0ZufTIciwh7cG8ExgxqYgeW3Uoe0PJwh1KFskAewFOVnYw0igenNF5JzBikg2cquf2uciVMXbsWO8IhSuULNyhZJEM8G4kBoSORMAVNTwowpORWLZsmcmFaUkCnw7C17N73om0gpKFO5QskgGZrW2VcwLFMrKGJyrPuHHjErmSZwZ1KKtDycIdShbJgI5EJ8J1u2PHjt7W9AeeoK+88kqiNHws3aZUuFiRMpQs3KFkkQwsWdgkvhnh/UhYvFWFaESMsvpBOQFF6qFk4Q4li2RAXkxS39GRiNJMz+QwuGwPGTJEChYs6BMFmbioaq6IHZQs3KFkkQwYsMR+2OLFNHwYFi5c6B2RNiAAjMpfGC/tfQlUI4JVEVsoWbhDycIBVAYjRZ4duARopZXXJqHjQVdyig8R66Eu22kDJQt3KFk4ggAx0ubZQYzL91tvvRWz6mU4UvXo0cOscNh7QEqUV1SkHZQs3KFkEQHwscBuYdPSkR+C9Hnr16/3jogOENFTTz2VKJM4ruRpre4olCwigZJFhMCOgZExGLRFNTEiUaMBiXKDvhOoHUgsGV3CMKdAycIdShZRgnqg5Imwg5yVCgoGuQLvUDpnsGZH8eLFjX1EkX5QsnCHkkUqQLbuevXq+YOdymUYI6lMlhzmz59vanZY71Bas2bNjCu3In2hZOEOJYtUghIAXbt29auQESKOAxeV2UOBCkOmb6qZWZLgvAEDBoQth6hIeyhZuEPJIgagaNDHH3+ciAQKFSokTZs2NQloiFylOnmlSpV84yitTJkyMmPGDO8qioyAkoU7lCxiCFYvqlat6pMB7aKLLjLlC4PVyWmtW7c2zleR4Ogf22TxhMHSq88QGTl9ieTsih+xgZKFO5QsYgwiU6llygqHzV4VbLiP4xUacYKa/etlznsdpHXPWbJ86VTp3+l5eWlQnOS8goOxhZKFO5Qs0gAk9SUNH2pGkCgaNWoUZfbvv2Rb3LvSvEpteWMRBYsOyXfvvyC1qjSRMRuOH6GIDkoW7lCyiCFYDl2wYIGxU1SrVs34TNAJUUWoChZ1uruj22VBvwZS4s7mMvXg8U075w+SumXulKajwpVNVrhCycIdShYxAin42rZta/JgEqFqpQmCwapXr27ctkmHF0lqfx8H1sn0NhUlf+n28mVCgkl2s2fhCKlXppDU7LbAO0gRDZQs3KFkESPg8l2+fHmfJIINt3AS/hL3QQo8VkciiinZv0YmtiguuQvUkA5D3pWh7w2Tdzo+I3fdeJNU7zrnmJKiiBZKFu5QsogBKO5TuXJlv6YokkXRokWNkRMVJNTQyfLp3XffbZZbnTrngfUyrU15uaZ4K5kSHy/x8btlw8y+8uCdN8mDPb/zDlJEAyULdyhZpAKHDx82Lt7BOJHChQvLxIkTTUEiGsFncXFxxs8CggjmqOBvAsZ++eUX74pJ4OhOWTSksZQs2lDGer5bv8zpKXVKl5aXpmgMSWqgZOEOJYsogecmnStXrlymoxExSvbvcJ6bgBUSDJzUHG3cuLGROCxpPPbYY7Jx40bvyHA4KjuWjpaXa1eVlybukH37t8rMfi3l8ce7yFwtEZIqKFm4Q8kiCixZskRq1qzpD3bsEZHUOsVACWlQ+tBeA6mD6yaJo7tl2aRe8kLLrjJwyDvy2n+6yNAF8cdoRJEaKFm4Q8kiAhDbMWbMGClQoIA/yG+77TZT3CeaVQ58LoK+GBhIWX5NGgmy95e18uPSlbJlT4ISRQygZOEOJQtHoHZQGtDaHCg6RF7Mn376yTsiOuDxSV5PrkkU6quvvhrd8qoiKihZuEPJwgEUGX7kkUdMh6IRKdqpUyfZt2+fd0TqQKg7Vc+5NraMRYsWeXsUaQ0lC3coWSQDoknHjx8vt9xyi08Ut956a0RJblxB9i3C27kHBs+MqFGSE6Fk4Q4liyRAWrvXX3/dDynHVwJ1IbrYjpQBMVF0mXthME2teqNwAyUqLVlQdQ67lCI8siRZbNq0yWSV+uOPtIm5ZJmzS5cupgPRCDFv37599LEdjujfv78plYjH5yeffOJtjRy8F1zLjxwh6OwEKIPITBor9Sk7gH6UP39+851RLSFtRXhkGbJAPCQpbs+ePaVKlSpGNXjppZdk4MCBJvtUrBrX69atm+9ohdGRGZ/tsb6Xbf369TP/4qBFqUTuScnC4L7QRmAaiX1D95N1q02bNtKgQQPzOyAgjqFhkCVEvmXLlua4pK6dUxrvgHiePHnymG9NciL6F9vDHR9t69u3r7kmtWmzMlFnGbJALA96SmrTltXa2WefbbK5Z1VkGbJAqrDekrTcuXMnyoxNo/APqwrXXHNNongMYjXYRiAXWbiDUaG0vHnzmn358uUz+iukxNKo3c997X728TcSgN1vr8/+G2+8US677DJ/H40VDruf+9t8nbZdccUV5jwS/ga3B58LiYAWfC5iUdjPdkTp4PtBOmE75wbrkZCxi22hjd/Nuwu+N9Qh9vHs7At6ndIo62jP5/7cL5gRjGvZ38174+/g8xPCH3yvqHvB98r7YD+N/aHvlfeGzwv/BrfznPa98Lus4ZjG89lnotnzg8mT6Vf2vuwP9104174zbEx2H2qkvXa4d4LUl1WRJcgC/bthw4b+CycFf6tWrUwH4v90SsRuYjJotWvX9j8QgxARkEjPuXPnGtuDJRnOQyTnHMoGIiZOnTrVqAAMFI6hw/Tp08fsI18mMwP3t52rWLFipoDx559/bsoDoKrgqEWnYT9p9lg9YT/3x4+CjsQ+juF38WyoFUFnL1zHKZFon2vKlCmmIpolBDrsG2+8Yc7lGKJZbael81PdjEA1VDY7WFjJQSzmeBq/lcb1uc6TTz7pvzeIl3gW9vO7UGmsIRDy4X1PnjzZfz7+JbmPJUIGNr+V6/LbKXHwwAMP+H4q5CsdNGiQ2c97JXtYyZIl/WA8nNUI6+e6s2bNMs8N0bKPxrvg2YYOHWp+l91+//33y8iRI801J02aZN6jNVIzePmW9p3NnDnTrELhSWvvSw0YHO/4TfxuVD3bz2hPPPGE31+4Bysolqx4d1S4T+qd8E5ZXcuqyPRkQVr9f//73/7HYqAyYNG/+T+Dlg8CoQAGKzkl2Aer01EtMO5ZF2tmMGweBIMF8f777/s6LKTCgLPAWYpU/5ZIsJuQ1t+CwDEGrZ0d6XjBPJuEsbONfTSem/vj8GV/D40OGVpkiOeyM1zoc0EqtkNzjC15SEe/4IILzHaqseNinhQwqNqZm849bNgwb4/I5s2bDemwj99GFbZQ3Rsyg5g5hvcHgQYBYdvn51mDiYoxxLZr186XaqgrGyyLwHvlnbCPBlHwvTHk8g7tdiJ/V65c6Z0lMmLECL+oNQQdWpOF81u0aOGT6T333CPr1q3z9ooJ8LMOczSqxgW/C+75EBz76BOspuDKbwGZ2QkAaYd3gvE8qyJTkwVEQSeyH4siPBAFM7/dRtp9vCABH8OKyoi048aNM9vB4sWLTdi4Pa958+Yn5ZQgw5WdBRhkw4cP9/Ycd/XGg9OK0cxSs2fP9vaKKVxcv359//p0ouXLl3t7RVatWpWIKOj8pP/fvn17ot9D52TVwoLOxXPZ34X6ZQcyg4xZjmdhH88OWQKMajZTF6JycHCGAuKx12cWxvBpQYCbJQpa3bp1TyrSzAxqyQrpBikpOCh4JvteQwct7xVpz87sRO0SpWvB++Eb2/sjqUFefDukG7u9VKlSZjKwgCjsjA8JIrkEwbunUr29L+kEWEa1+Pnnn42Eaq/P38HoYBz1bCU5+gQSRXC1DKKw74S+1KtXr5MmpqyGTE0W5ImwH4vU+nQAOqvdRqEeW2cU0RMxj+10Dga6dZvGN6Js2bL+ecz+zFYWHEd8h5VIMEQFBwwdEyu5FdEhImZSCzo06ozteEWKFEnUcXnG4AyFOE7Hgwx5FrsdMgnWTYUMEJVROdiPbYTVDQuMvhRPZh+DvHv37uYcPEJtWYJwM2oQqBJ29mV2RH2xYMAEJR5+AwM1iDlz5hiphf28N2qoWF8F3ivfzL5XvgvEZ78Ly5SoKnZmZxbmehah6iffEAJGImCg2+133HGHyaxuwe9FfWQfUg5STXAZmUHNMqm9LwRFzlQLyBApwl4fiWXDhhPJTokstv0JybZJkyaJYnqYRKzKZN9JdnD2ytRkwQeCEJhNELUZDPYDEnRlRU7co+3HYQZDv7XiIB0zODPVqVPHDO4g8NuwszOicHDAAGwN1ihKhw+qAGDw4MG+CI29gjycQUAk9v4QAvcDDBS7nZkxOLMBOjiqBPvtc9mBBpCWIAUkCKQe63eyZcsWU7OkYsWKRv9OCpABz8v16fRcw743Bjz6uH0+nhsSCgKpo0KFCmY/Aw8pMKieICFYMZzvAgEHxXS+qX2vEDDEZcFxSCj2/kiFNioXArLb+e5BVRD1xRIo1+adhfpOMJEEbSfYJix4v0Efm9KlSyeSEJFIghIkEmJQEqTP2nfCN6OQdnbxa8nUZMGH40MwmzFwEFf5CHTc4MCaN2+eEbXpkEgAQVdprgFBMOsjSobOjIDBy0yECB0cMBbMdnQuZmlsB6Fg5uL6zFDBjmeBysN+CC7omYmOS4dCvUoqHgSpiA6HihUqxvI7mcWQikIdxiCClOqSoL4hsTH7MdCD742/qW2CMTP0fVusWbPGkBxGV461diMLVrDsd0EMD3Vhx7aCWsXqRCgB8w0wqiLiY0TmGwP6AUZNyI33jeQVBASKwRMCZUYPl74QcodIsLGgLgRBf6Gf8V1QJZFug8BmweTDc4XLQwKhYpxFSuOdJB9FnLWQ6Q2cQWB8QvdE/w+CTojFHOt3aIcEdHQ6JjNuONABOR/9P9z5zAzMguGIAJANi84eFIWDYNBiawmdmXkeDItp5ULuAogQC72VSoJgluS5k0roA3khvvMbQokC8C4ZzEgM4dyomaV550HbTxBMFLzX0DwfSIY8V7j3zX2QBPleSeU5ZQCz3xJQKJhQuG84goRMkDS4f7iJB1sNkiXvJLtVws9SZKFQKDIOShYKhcIJShYKhcIJShYKhcIJShYKhcIJShYKhcIJShYKhcIJShYKhcIJShYKhcIJShYKhcIJShYKhcIJShYKhcIBIv8HxD58OuBsCJ4AAAAASUVORK5CYII=)
physics-General
physics-
A billiard ball of radius R is struck by a cue as shown. Assume that the impulse given by the cue is purely horizontal. For the ball to roll without any initial slippage
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aJKr8riTLRQth0j1Ut2EddZUZLz4i274TbOiQTJQvFKSVpJAEdX8e57ZhOBK2/maRRf8kaHWle9WQjhTAxKSB8MpUGnpkarJaw6sbDsA6tb2FkgRI3CQkDXpmNAmnFxOMRQpUJhBAJq2wvgZM8MTeZQBI25UlcTlYO92+ihFBGPOKdMvi22ba3HYORKDEUc5Ntj/1QSSFhI57rPcFq++54GefS+t0wHRvnyDmZLqQQJiYcyjYUvbcNgqkkcWORyroa8Ep21NdxixdVCMUXWYSsKC4xERSn67VgrJfAKY0hhhNpERL0iBOKeZ500knVems7BiOREIoZWgcLkHXm2BB5n4nfyaS3fXe8JIJuUoRQ6IAVzR2eTqQQJiYcYmPEwYXugm8bDFNB4nb66acPJ0rUMHKtIzs5HiFUY0jYCJisbRvE4dRGqvkjLJMhhDLiLFHt/tuOwUhsCqEaW+fKMRCrE4O0rZMphLwEsUiJGq76dCOFMDFhEN+RJDCYiUe4xNNFA1tQPoRQRpLlMV6LkNWk6FrihdsLbTEt4QHiq3DdcpMlhLZlPEIozilm5xiok1TmpHB/soQwOHfu3PqbMNXJkV6kECbGhaYAmKWj0Ncd3+wFr20DYKrYK4REabxCGDFCNJNHCUpMazOYHY84JrKhhIzwSZj0q0UYQohTJYSOM0vQsXPcUggTA40Y9DqcmB5mShrE7Iy2QTBVJG5N15i4EMKJcI1ZYaZfEitlNL1QFEwo/WY8zGkyhFDiRNJm0ITQtUEIm9MCpxMphIlx47zzzquDNwphCU0/CKFt4H6xBA14dWoG+0RYhOJqEiZq8SRMlMqICQoHWFZTDckZs1ckBCYya9wUQg0uJITsV9sxaKPzIjkSQmiaoBihdbih+cy2Kstp+/5Y6feidMf7YAphYuARlqCLW5mIpqTwz3/+s9YKhqi0DYypJEuJZWZws3S477adEC5KQXVYhFFQjUpPzBdWPygDqjkFgVNDqPnCRNYRmrJo+qD9QR2LWHdt+z4S7adic9tAWLfbbrt6w7DvZpz4DfFPD6tyftvWMVqG4FmPYm7JM/S5G1UKYWLg4YLWJMC0MlPqDLDIEveDCCJLx5S3qLsz6MFDj8yvlfEliKwfcU3LG7SEgRASDO6+VzMhWL0hiNGlBs0wUaCs2YKMud+UPY8uQ9xjTRdCDKPpgkRKiGhTCHXtkdxh8enGw703xY5waStmX+yTdXNhCYv9dQ7sA3FvoxsDnnvuuXUmiRiqV+VOhMr+EX31lva793iOhbZJWQ/r083A76gbbBbW90uD2BTCxJgQQW0XsZZd0T6emBA/F3+wbXBMNbmqBEwjCOLBsopO0sRIezKWUbQmQ23UWILxBDvJCP0oWW0KmKPn47rrrlt7O6IkETFTM6hQmSASQRYi1/zRQ30bCRsrT21jtBlj8YmvNoXQcrbZ3GKusFigaYJEMETdq9IcHadZXSwsAmcbCCyLs0mf2T/twSxHMB0f340bmHWwjsOK6z2eY6F1Wp+wgN8eqcyoH5BCmBg1wh12QZtCRkzCtVGM2y/i16SBaIYL0SIeo6X5ybK+BEOjh7Zl2qivIAuZmLL+zNJoW66X3F1iysIWcyR+bcu10e+x9gibec5ty/SSWPtOCOBkkOUpfsny9VsQ11C/IYUwMSro+OMiJnxcO5aUucSysESQC9SPQoiSAGaBaNrKiiI6LDWxOVZcdPo2vYx1Z7qX2RrihCxCc4lZghqmRhdvrmuze7d4nuSIm0NYhGoHLc/iYwHqwNPszB3duMX+FF4TQb/vOyzFsARtbzSWDWvQZ2KztlWmN6w4zy/mamv4ynK1n9Zn2+ynfWSdiXv6TtvxGi+552KvMS+btwDTXSKzIKQQJsYEiQECIiYoznPhhRf2nTvcSy4ekZBYCBExH9hUQIkGlhTL0faLk2nI6lU2XJyMVSNuRmSaiRLiGg1Zoxeh6WrRj5ALyk1VOqO7uDZThFJ7MK44QepNlpiSR7iILHeaG82FJqCEPCw6Qkeo7RvhiX31XoiCNWu/xDF9n6CKj7IAw+2drPPlxuO42BfhBehXSzCQQphYIKJIGLiLrBCDCQTpCaGLv19FEG0bi1VCI1xVFhiRI4asF24cC8m+caUjayxJwnpCCQSWIkun2Z1a55Zmd+pmY9ZIlBBDQiSmJ2vM3RavI85a9/dmjaNVv1gha5L1GNtuCtzBBx9ckzvNOF6cg7DIJD7EJUM8xReV84gLNsVzIklkHTP7JDYL/S6CkEKYWCDiIiYiLBWDDbjKXOMQwn6ngW/bWWEhDLKxssPc+9GWz4S7F1ZhszHrZHSo9jwX1mBkilG803EfSczsi+3nasd3mrRdLOFFTYY4jm20PiLrmOEgIYUw0QoCGDEdloW4mEny4HMcJCHEGKSSEiEK4mZcYdatgdxPQqh8hjXYbLtFHJWh2Jfe/SNGLDICySKLfeylOkEPJ2MFj1YMQ+wiq9yk37Qu7rgwApdceRIMgjUIKYSJVoQIypoSQfExiAscBk0I0YDlEkdrLpRAIHzq3Qz2fhBC5TQSHM3sscSKWS0ysG37RgCJEvc3vtPLEFTP+lEET8hGkzl2XLwSQjHAIEH23BR1mGbTWKfawLhZphAmBhIuXm4vGFSsJ4MdXNTxPxhEITToWSymyMUcZCRAiov9j1UYQhgxwskWQmIsASJrLA4rUUKsYvtsq3jkSH0H7ReRN20upt+1kRCGGCrO1k5srDNTwjpEv+tVvNMT+ogihBAOClIIE/MhhI6lwEoxh5YV2HZRswIGTQiDBE/CQ0Y1RILYyPRylYnhd7/73SkTQsdaiY5Mq3KbZkxQtlgWdqS6P0LEOuPuErf43mhIcDWmcNMbjWWIfs/x8V6rsZ122qlmwk2vHFSkECbuAnd1MSYDmQiqHdRNRTOFJmUtB00Im5YMC4qwRVOGoNpCGWGCSAy50pMlhLLGMcWOO2waXVhsaNuU6RDukYSKu8qC7d0PtL542BORFB/sXca5ZmmOVgjFFS3PLRavZKk2nz8yiEghTMwH8R1T0Ew/MxeX22SgsRia9JmBQ1TaBsugkMAQEWUuprOFOGiBT7h0mGERehaJ8plozDpeIYypffoZsgSVuTRnobAItcJSy7ggEUTCJFHBPVWcbXaLUh01hMpswlWOkhZlUH5bDHKVVVapU+B4AG3r7iVLkPUJZrEQctMtBx0phIlhcG0MVpbKNddcU2677bY6AF38YUk1GZ+3DZhBooFNCFiB4nQhRihZoVyFqIkrsgq1sw8x9JAjbuuBBx54FyHUeYYQxiM9FVWLoxFDx1g80AyT5lPpULySuEp+2La2bW7SOSCGHtdJtNVB2h9uq+2M9ZqFwnJjCbuRKRoXBx2LVe+3eAKOhSQai3kmIIUwMQyzRpSTqKsDLrHBOBPEbmEk+GJwxEH3GCLVnMmB4ommrRE4YmcWiUJqrqtHlhIvzVoJogepW4aAsgZZ2WoYJUXUBvbG8pTIqPvTsYaFGnV+Yzn2zhV31av9YUmyaOM3xCP933oxLP2x/I7lucMy3MIFMEhJkZGQQpiocUDuEkvEgAAZYhZhV4QQ7aeBzgrzXskQC045SzwgPiiOx5U2Q0WML9pqsfQIHtHxOcFg9RG65vdRf0RzlvVLJFgsM79NoNq2b6xckBC2LT8auh6ECIQKYNCywyMhhTBR6+ZYLOJHrMDoF6eD81itkkFn7KsBT5RYZlzOWbNm1WlxZnk0C5zHQrE631155ZVrjFCMkWsp8YB+s3d7xsPJEELHxzpC/AalTnBhSCHsKOICNtj1s3ORu8C5VBGYd9F3SQRHIisxhIolJO4mOaLfoJiq5IQCbTRtL6gMxpxhzRa09OcyK9kxQ4fVZ53WPVnHeCKE0LZZPm6I4sLW2y8NVScKKYQdRIigQa2voNkjkTUM8Qv2DowukxASEseKiIkpSjywGFnTaIpZUOKCxWc59B3fJX4Tbf21cbxC6PxHdYBSoviem2cKYWKgESJoYJo1YkqU972DIDl6EgsC10uCQfTavjMVJISyuyGEOorbntEIIctPwbzqAY8YEBrQqUdTXu32xZVnilsMKYQdQsR1WDGyw7vvvnstmWElpPU380iIlfvEVD1lOwQ6Qh8LoxixmUXqD71vYiaJIKQQdgRx4bqjy2aKb/msSyUyXSPLTyZaSY+Cca76aDPSRHS//farU/9YkTATssMjIYWwA4gL2LQ4wX0zG5TGwI033phCOMPJCuQmt4mg896kzyJppsDcK8w0C7AXKYQdAUtQrMeUKM8YCZhDnELYTTrn4sPcXtcEmmcuwaKLTRTWz3QRhBTCGQyWoBggS1CzAPNLZS5ZgwTQXGIXu7u+4HjbYEnOXDrn6kWbMOVOkbcHQsFMdoebSCGcwdBSS2KE0Jm2xQKIYHmTbYMkOfNJCD2EK8Az0AvRM1i6hhTCGQh3ceTSEEKdSVzk4QI32TZAkt0gIYxHbUqOmF/NJQ5XuAsucSCFcAYimqsq9FXYSwDbBkKy25RRVhMoTKKFlwarXRK/JlIIZxDCEoRzzjmnNvv0+El3+7aBkOw2lddoy2UKoA7T0Wa/i0ghnEEIEZQQ0UlGU05NA9IFTgabIRF9BT1vWZ9CZVRdRgrhDIMpUWYCKKDl5hDCQWunn5wcSoyxAoVK1AqKHXsgfRRMdxkphDMIMoD64Wm9fuuttw5njXUsbhsYye7QzdC1wPIzT5gAbrPNNsO1gl1HCuEMgXpBz6nYaqut6oUOxNDFnxZh0s2QKwxnnXVWrRX0UKrE/5FCOAMgNmj6lPqvuXPnliuuuKIGwc0a4ApljDDpZqhUxgPrPWvEk/ji2kmkEM4IaBIqO3zllVfW+I+LPpgzRpIoDqj/pCayxxxzTL1umlUGXUcK4YBDu3cXNzdnOnvfJfubvIMDDjig7LHHHjWJFkz8HymEAwzuDRHUGZkrnEmRJPICwiOQKVZHKjzivbZrBDCK7hP/RwrhgMJzRtZee+1y5pln1r/NIXbhtw2MZHdI8AifG+NVV11VQyX6EWqm0OWC6YUhhXCAEK6Mp855xOThhx9e7+xqByVLMinSPTrnvVRGFVbfDjvsUGeNNJsrJO6KFMIBQYigO7xnjRxyyCH1b73kCKMB0DZQkjOTcb6jg1BQlYBni4CYoOcqX3311fXvxMhIIRwAhAjqI3jooYeWPffcszZa5eqYOZKZ4W4xLD9lUmJ++k0G/Q1KqbTU0n4NMju8YKQQ9jlCBN3VxXpc2GJA7v4xINoGS3Jm081PSKQNp59+elljjTXmE8G4jhLtSCHsY8TFK9az7bbb1ilRYoHZVivpBtgmhNquKZjWkTwxeqQQ9imi2NWDtHfdddfaUstEedZgusLJNiH0ebNgOjF6pBD2Ofbee++y3nrr1VigOcQxCJqDItk99gqhcpkNNtig7LXXXkOfJMaCFMI+AyswmibMmjWrBrw9ihEkRyI22DY4kt2ha2DevHn1uiCInlW94447Vg8CMiY4NqQQ9hmi4n/27NllrbXWqrNGAimEyaBrQBMFlQTix1tvvfWwCCbGjhTCPkHEBMGskTXXXLNm/5roRyEUr0xOPdULmjmiYHrTTTet9aSJRUcKYZ+BuGy88cb1Idu9YC2aPK8Vvz6D003bwT3z8J+ZSn0ePeBIZx/9/NTu9QNty2677VY22WSTFMEJQAphn0CnYE0UVl999bLRRhvV5pmnnnpqOeWUU8ppp51WrUPu8kEHHVTrCb1OJ20D6n5j+2Yy58yZUx943i+0PbvssktZZ5116vzyxPiRQtgncFc/8cQTy1FHHVUH38knn1z/7uUJJ5xQqU5suhnb0radM4l6PfYTieGxxx5bzj///KGrJzFepBAmEonOI4UwkUh0HimEiUSi80ghTCQSnUcKYSKR6DxSCBOJROeRQphIJDqPFMJEItF5pBAmEonOI4UwkUh0HimEiUSi80ghTCQSnUcKYSKR6DhK+R/fKjhOvRAf+gAAAABJRU5ErkJggg==)
A billiard ball of radius R is struck by a cue as shown. Assume that the impulse given by the cue is purely horizontal. For the ball to roll without any initial slippage
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUIAAAB6CAYAAADK81roAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAFxEAABcRAcom8z8AAB/QSURBVHhe7d0JuG1jGQdwUUiRFBFpHog0D4QUDZTmhJJKJU0SEomohBLJGDelbiiusTlD042KlDRoMFxDE0UZUn35fZ33WHdb595z7pn2Puv9P8//2fvss/baa/r+652+dy1WEolEouNIIUwkEp1HCmEikeg8UggTiUTnkUKYSCQ6jxTCRCLReaQQJhKJziOFMJFIdB4phIlEovNIIUwkEp1HCmEikeg8UggTiUTnkUKYSCQ6jxTCRCLReaQQJhKJziOFMJFIdB4phIlEovNIIUwkEp1HCmEikeg8UggTiUTnkUKYSCQ6jxTCxLTjv//9b+XC8O9//7v85z//GforkZg4pBAmph1E8K9//Wv57W9/W6655ppy9dVXl9///vfD/N3vfleuuuqqcsstt6QQJiYFKYSJvgAhPPHEE8vjH//4svHGG5dDDjmkHH744eWII44o22+/fdlggw3K5ZdfPrR0YiTcfvvt5bDDDisvetGLyimnnDIqSzuRQpjoA8Rgvemmm8qDHvSgsuGGG9b3XOHbbrut/PGPfyzveMc7yi9+8Yu6XGJk/PSnPy33v//9y2KLLVZe8pKXlH/9619D/0ksCCmEiWlHCKFB++hHP7o873nPqwIIxBAuvfTScu2119b3iZFx3nnnlbvd7W7DQhjHMbFgpBAm+gbcOkL43Oc+d3gAE8mbb765/k0oQxgT7fj+979fllhiiSqEL3vZy9IiHCVSCBN9A0L4qEc9qrzgBS8Y+qSU6667rsyZM6e6ykQxkyULRgrhoiGFMNE3MGjXXnvtyuOOO66cfPLJZeutty6vfOUr0xIcJVIIFw0phIm+AYuQCD75yU8uX/3qV8u3v/3tmgHdZZddyq233jq0VGJBSCFcNKQQJvoG4RpvuummQ5/8H7/85S/LP/7xj6G/EgtCCuGiYfqE8D83lYvOPb0cdejB5aBPfLIcP+fscu0tQ/9LdBLNZElYgJFRjtepRMQkvfb+vr///Oc/lz/96U/lnHPOKV/+8pdr3V6TX/rSl8o3vvGNMm/evPKXv/yl1b1vW/fCEHFSWXThg5/97GfDyaWLLrqo3P3ud69CuMUWW9TPQMLpkksuqfFWhetj/c2ZjukRwn9cXvZ/w4ZluSUXqycMl11ji3LBP4f+n+gkDPDHPOYxtXym3ywZwvHzn/+8uuof+tCHyutf//qy+uqr15q9ZZZZpiy++OJ3oTKWpZZaqtzvfvcrq622WhWmffbZpxx11FHlwgsvXOTSlhCxbbfdtlp/1r/++uuXD3zgA2W//fYr97znPeuY2mijjaogv//97y/rrbdeue9971tF0nu1mYk7MS1CeOUZ+5YVCeDSDy07fOjIMnv2F8tp3/xxuWno/4lugvv7wAc+sDzjGc+oWeKpBiHuzUr/4Q9/qNZeCF/cuMfLlVdeuQrjZz7zmWqhNRFW4sKstve85z2t6yZ2IcZt/4+C9cSdmBYhvOS4d5Wl7zghKz53z5JGYMKA52Z+/vOfL09/+tOrdcOS8dlUIkSQG3v22WeXt73tbeURj3jEsKuJ3j/4wQ8etly32WabmszZY4896uu73/3uOgvmrW99a3nzm99cXvOa19TlHve4x5VVVlmlrLDCCvOtj9XIWvS9733ve3WedaBXlHthWuKzn/3s4XUtiPGb66yzTrVGE/NjSoXw1qsuKp8/9siyy5brl7vfcVJWWGfzcsBhR5Yjjj6pXPqnlMQu429/+1udS3z99ddXIfL+xhtvHPrv5KDN6vrKV75SXcd73ete8wnJIx/5yPKqV72q7LbbbuWYY44pxx57bJk1a1Y5+uij63zoT37yk+XjH/94+ehHP1r23Xffsueee9Zld9555/Kud72riurrXve68upXv7o85znPqULKpW3+xtJLL12e9KQn1TnXEU9cmGWoIYX52c31jEQiyL0H8djEnZhSIbz+7IPLmqveryy7zFLlbnecmCWWXKYsv/wK5b4rrVuO++l1Q0slElMDYhMiQ1C4v8suu+ywcJj3vMkmm9TY2xe+8IVy1llnlTPOOKOccMIJ5XOf+1z59Kc/XRtDEMGPfexjVQTFD/faa6/yvve9r1qIIYIaR7zxjW+sYsiK5BbLjj/hCU+o4YCm8IorsiSjycRIQhifKzMKtz2m1wUjg8wS/drXvlaXT9wVUyqEt98wr/zkgrnl6J1fUpa84+SstN7ry6nn/qB8f+5F5bqbck5kYuoQIkIMZXcJUlM8dMBhmSlH4Sara5Sh/eIXv1hmz55dC77HKoRveMMbymtf+9qy5ZZblle84hXlxS9+cdlss83K85///LLuuuvWuGEIF7L0hAvCRe4VRJ+HZUec73GPewx/t0kJHfsCbZnrxDTFCC+e9Y6y1B0naLXN9ynXD32WSEwVQgz+/ve/V5FigYVoPOxhD6stwAjgBRdcUK0t7jJLUNySELIOxyuEL3/5y8vmm29epxOyOmV4xUbVUXKRY3tYeO985ztHjJeGOBJFv+07kiQhqCxNwh0YybrsOqZFCH94xPZVCFfdbM8yLw3BxDSAILz3ve8dFhzCQZyIngJuIvid73ynfPOb35xPCIkKK220QkjEQgi53lxeQqjYmRCyBlmfMrl6LsqYP/axj63xw6Z1KPkSJUW9YuZv4o5vetObhr9DEJXrxDKJkTGwQqjg1h1dyUUyOVoqLFY6cuCBBw5bgkRHwkN3bJ2wCaEi5R/96Ee1WLophNxjr1zR448/vpa/SJioL/zEJz5R1/uRj3yk1u5JlIQQRhY5rEIxQu5xWIUSKMSQiyxhIsssdhjZXqJGXBcGlqOymrXWWqsK9EhIF3l+DKQQulhl8LgT7qbJRSe3rEtUyqLkpOkOP/OZz6yCRgxlfokZy46Vd9ppp9X44JlnnllF0P+VuoSVxwKLJAhrb6uttqrX5lve8pay6667lh133LG8/e1vrwkSM2aInd8jeE996lPrvGqxQFbgGmusUdZcc836twyv95IckQCxzR/84AfLqaeeWrclKMZpxkiz9IbY/+pXv6qirQ4yluXaK9PJqXfzYyCFUPwmLuJkclHZm2Fto5b3LEIJEwK53HLLtS7XSwkKAkkMCeZoi7FZfrLVIYQrrrhi6zK9nyF3m8ULLMOnPe1prcuZeUI4E3diIIXwhz/8YT2hyy+/fNlpp53qxcYdCLpjo0B1kIvizrzDDjvUu7nXoLs3tyW43XbbVXJj0FQmdNdHro27f7yK+QTVibEIUPso7o8LFHUMlikMGmQvfOELh19lEJVUIHeJ9YJhzbDguFDIqmERP+tZz5qP4kwsDoF3ZHkEDQxUtBzWCDcMn/jEJ9bMKWsEuWY6wXCxWCtBg5Plog7OvGBUYxeUbHj4wx9eHvrQh5aHPOQhtfg4SAzQQFdEzPVDhcYypg94wAMqV1pppSoAxITbyipCU8SccyRIqNzl3ve+d53mFpRsWHLJJStlUr1yMbEpfv62LuuQVGjSOvym880iFCuURLFfhMT/vfYyPleI7briHnOLHUfbZnt6GdsmJsjqc6wce8fZsbFMr2hbl+Nh29F7vxNztM1Ddu3FMs3jZR8Ie+JODLQQGjQyeEx+QeygYDYef/zx5bOf/Wwl10cRrAC3mA4eeeSRleI7n/rUp8qhhx5a6YLHgw46qLpCXCQ84IADyv7771+D4uZ0igV9+MMfrgHp4N57713rzhTUihOZcbD77rvX4lrBeQF0DMHmOoVoh1iHULMkMIQ63DBBd2wKM+rdR4zFnwgyhhh7JcYvfelLK4kyAW4KL3LfmsJLcAlsBPNDYMO9E9wnqviUpzxlWFiJKqumKahENFxAQipDSjyJhsEZ4okhmKuuumpliCWRdN6JZK9AxmAnZIQihAkJDEEhNiEqxEdxtPo6LiQXk+vounGtuIY0TSCEYoSsKNePc+48O8fidm7EzqHzFzdWAujziBE6f5IxjnW4546p4m3HzjELl9ixc7wcJzcWNw03BvsRQsjV9nySn/zkJzWW6b2n/AUkR8TQfe7/lvvxj39czj///Fqf6P9mptg30wi7jmkRwu8d/Np6Mu+1wc7lykUQwh/84AfDF0QyOR7KDpvBcuWVV9ZHh/7617+uD4kiIP7HcmomS4ilG28UVRNMTRTcQN043SwJZUy5i/KZSJa4gUWyxI2JJyBzHMkSNxbWeViEhJCF6GZA7EPE3XTGMwXR/rlpulmwdruOaRHCv1x2QTnhDsttzrkXl5sXPJ2yFS5W1sR97nOfynCXgmEhxGsvF/b/oP/3A8PyaVpAY+VI322udyLJpUW/4ViGNdZ0H8Naa6P/L2yZXsZ3er/LQmQRhoj4jLXJepNQuOKKK+qrNlXaWLGg3GyVz3zrW9+aTwgnq46Q1c3KFrroFULWsRACK7iZ5LH+G264oU5LNEVxNGQFIivQ963HvsmoS7BMF92MxDdtm7+nutxnWoRwvJD61wdOlkw7Ie+T/UXWinPDejLQuXdeo2xkPIwY7GgYsVsWlUFPnFlzegTaThUIptdNhBAKm4TbPBlCyD0W41t8KFkiTCD8Ym6zsIzXhdFywjRCM9x0NweWqe3ULALDgp0KCiOgEIIwj/CJrPhUYyCFMDE44HYZyFxGtXlcMmKzKPRdcS7iZL2jITEjUKxA4iHBxPoQTyPUrMHxCqEYMzFkEbIGlbh4FROOGOFEWYSsa7FN+0IQxaovvvjiGjcfDcUI1UZy+cWnrce5sa9ioETo61//+nDJ0GTSb/hd59QNk5UuRu4GtbDOOxONFMLEpIAbxtqSiFCfx3rXVQbDPVsUEq/LLrushkd+85vfLJQsPxaQAY8SHGBdnpA3HiEUG5RsYQ0SNoIWCSaJKr8riTLRQth0j1Ut2EddZUZLz4i274TbOiQTJQvFKSVpJAEdX8e57ZhOBK2/maRRf8kaHWle9WQjhTAxKSB8MpUGnpkarJaw6sbDsA6tb2FkgRI3CQkDXpmNAmnFxOMRQpUJhBAJq2wvgZM8MTeZQBI25UlcTlYO92+ihFBGPOKdMvi22ba3HYORKDEUc5Ntj/1QSSFhI57rPcFq++54GefS+t0wHRvnyDmZLqQQJiYcyjYUvbcNgqkkcWORyroa8Ep21NdxixdVCMUXWYSsKC4xERSn67VgrJfAKY0hhhNpERL0iBOKeZ500knVems7BiOREIoZWgcLkHXm2BB5n4nfyaS3fXe8JIJuUoRQ6IAVzR2eTqQQJiYcYmPEwYXugm8bDFNB4nb66acPJ0rUMHKtIzs5HiFUY0jYCJisbRvE4dRGqvkjLJMhhDLiLFHt/tuOwUhsCqEaW+fKMRCrE4O0rZMphLwEsUiJGq76dCOFMDFhEN+RJDCYiUe4xNNFA1tQPoRQRpLlMV6LkNWk6FrihdsLbTEt4QHiq3DdcpMlhLZlPEIozilm5xiok1TmpHB/soQwOHfu3PqbMNXJkV6kECbGhaYAmKWj0Ncd3+wFr20DYKrYK4REabxCGDFCNJNHCUpMazOYHY84JrKhhIzwSZj0q0UYQohTJYSOM0vQsXPcUggTA40Y9DqcmB5mShrE7Iy2QTBVJG5N15i4EMKJcI1ZYaZfEitlNL1QFEwo/WY8zGkyhFDiRNJm0ITQtUEIm9MCpxMphIlx47zzzquDNwphCU0/CKFt4H6xBA14dWoG+0RYhOJqEiZq8SRMlMqICQoHWFZTDckZs1ckBCYya9wUQg0uJITsV9sxaKPzIjkSQmiaoBihdbih+cy2Kstp+/5Y6feidMf7YAphYuARlqCLW5mIpqTwz3/+s9YKhqi0DYypJEuJZWZws3S477adEC5KQXVYhFFQjUpPzBdWPygDqjkFgVNDqPnCRNYRmrJo+qD9QR2LWHdt+z4S7adic9tAWLfbbrt6w7DvZpz4DfFPD6tyftvWMVqG4FmPYm7JM/S5G1UKYWLg4YLWJMC0MlPqDLDIEveDCCJLx5S3qLsz6MFDj8yvlfEliKwfcU3LG7SEgRASDO6+VzMhWL0hiNGlBs0wUaCs2YKMud+UPY8uQ9xjTRdCDKPpgkRKiGhTCHXtkdxh8enGw703xY5waStmX+yTdXNhCYv9dQ7sA3FvoxsDnnvuuXUmiRiqV+VOhMr+EX31lva793iOhbZJWQ/r083A76gbbBbW90uD2BTCxJgQQW0XsZZd0T6emBA/F3+wbXBMNbmqBEwjCOLBsopO0sRIezKWUbQmQ23UWILxBDvJCP0oWW0KmKPn47rrrlt7O6IkETFTM6hQmSASQRYi1/zRQ30bCRsrT21jtBlj8YmvNoXQcrbZ3GKusFigaYJEMETdq9IcHadZXSwsAmcbCCyLs0mf2T/twSxHMB0f340bmHWwjsOK6z2eY6F1Wp+wgN8eqcyoH5BCmBg1wh12QZtCRkzCtVGM2y/i16SBaIYL0SIeo6X5ybK+BEOjh7Zl2qivIAuZmLL+zNJoW66X3F1iysIWcyR+bcu10e+x9gibec5ty/SSWPtOCOBkkOUpfsny9VsQ11C/IYUwMSro+OMiJnxcO5aUucSysESQC9SPQoiSAGaBaNrKiiI6LDWxOVZcdPo2vYx1Z7qX2RrihCxCc4lZghqmRhdvrmuze7d4nuSIm0NYhGoHLc/iYwHqwNPszB3duMX+FF4TQb/vOyzFsARtbzSWDWvQZ2KztlWmN6w4zy/mamv4ynK1n9Zn2+ynfWSdiXv6TtvxGi+552KvMS+btwDTXSKzIKQQJsYEiQECIiYoznPhhRf2nTvcSy4ekZBYCBExH9hUQIkGlhTL0faLk2nI6lU2XJyMVSNuRmSaiRLiGg1Zoxeh6WrRj5ALyk1VOqO7uDZThFJ7MK44QepNlpiSR7iILHeaG82FJqCEPCw6Qkeo7RvhiX31XoiCNWu/xDF9n6CKj7IAw+2drPPlxuO42BfhBehXSzCQQphYIKJIGLiLrBCDCQTpCaGLv19FEG0bi1VCI1xVFhiRI4asF24cC8m+caUjayxJwnpCCQSWIkun2Z1a55Zmd+pmY9ZIlBBDQiSmJ2vM3RavI85a9/dmjaNVv1gha5L1GNtuCtzBBx9ckzvNOF6cg7DIJD7EJUM8xReV84gLNsVzIklkHTP7JDYL/S6CkEKYWCDiIiYiLBWDDbjKXOMQwn6ngW/bWWEhDLKxssPc+9GWz4S7F1ZhszHrZHSo9jwX1mBkilG803EfSczsi+3nasd3mrRdLOFFTYY4jm20PiLrmOEgIYUw0QoCGDEdloW4mEny4HMcJCHEGKSSEiEK4mZcYdatgdxPQqh8hjXYbLtFHJWh2Jfe/SNGLDICySKLfeylOkEPJ2MFj1YMQ+wiq9yk37Qu7rgwApdceRIMgjUIKYSJVoQIypoSQfExiAscBk0I0YDlEkdrLpRAIHzq3Qz2fhBC5TQSHM3sscSKWS0ysG37RgCJEvc3vtPLEFTP+lEET8hGkzl2XLwSQjHAIEH23BR1mGbTWKfawLhZphAmBhIuXm4vGFSsJ4MdXNTxPxhEITToWSymyMUcZCRAiov9j1UYQhgxwskWQmIsASJrLA4rUUKsYvtsq3jkSH0H7ReRN20upt+1kRCGGCrO1k5srDNTwjpEv+tVvNMT+ogihBAOClIIE/MhhI6lwEoxh5YV2HZRswIGTQiDBE/CQ0Y1RILYyPRylYnhd7/73SkTQsdaiY5Mq3KbZkxQtlgWdqS6P0LEOuPuErf43mhIcDWmcNMbjWWIfs/x8V6rsZ122qlmwk2vHFSkECbuAnd1MSYDmQiqHdRNRTOFJmUtB00Im5YMC4qwRVOGoNpCGWGCSAy50pMlhLLGMcWOO2waXVhsaNuU6RDukYSKu8qC7d0PtL542BORFB/sXca5ZmmOVgjFFS3PLRavZKk2nz8yiEghTMwH8R1T0Ew/MxeX22SgsRia9JmBQ1TaBsugkMAQEWUuprOFOGiBT7h0mGERehaJ8plozDpeIYypffoZsgSVuTRnobAItcJSy7ggEUTCJFHBPVWcbXaLUh01hMpswlWOkhZlUH5bDHKVVVapU+B4AG3r7iVLkPUJZrEQctMtBx0phIlhcG0MVpbKNddcU2677bY6AF38YUk1GZ+3DZhBooFNCFiB4nQhRihZoVyFqIkrsgq1sw8x9JAjbuuBBx54FyHUeYYQxiM9FVWLoxFDx1g80AyT5lPpULySuEp+2La2bW7SOSCGHtdJtNVB2h9uq+2M9ZqFwnJjCbuRKRoXBx2LVe+3eAKOhSQai3kmIIUwMQyzRpSTqKsDLrHBOBPEbmEk+GJwxEH3GCLVnMmB4ommrRE4YmcWiUJqrqtHlhIvzVoJogepW4aAsgZZ2WoYJUXUBvbG8pTIqPvTsYaFGnV+Yzn2zhV31av9YUmyaOM3xCP933oxLP2x/I7lucMy3MIFMEhJkZGQQpiocUDuEkvEgAAZYhZhV4QQ7aeBzgrzXskQC045SzwgPiiOx5U2Q0WML9pqsfQIHtHxOcFg9RG65vdRf0RzlvVLJFgsM79NoNq2b6xckBC2LT8auh6ECIQKYNCywyMhhTBR6+ZYLOJHrMDoF6eD81itkkFn7KsBT5RYZlzOWbNm1WlxZnk0C5zHQrE631155ZVrjFCMkWsp8YB+s3d7xsPJEELHxzpC/AalTnBhSCHsKOICNtj1s3ORu8C5VBGYd9F3SQRHIisxhIolJO4mOaLfoJiq5IQCbTRtL6gMxpxhzRa09OcyK9kxQ4fVZ53WPVnHeCKE0LZZPm6I4sLW2y8NVScKKYQdRIigQa2voNkjkTUM8Qv2DowukxASEseKiIkpSjywGFnTaIpZUOKCxWc59B3fJX4Tbf21cbxC6PxHdYBSoviem2cKYWKgESJoYJo1YkqU972DIDl6EgsC10uCQfTavjMVJISyuyGEOorbntEIIctPwbzqAY8YEBrQqUdTXu32xZVnilsMKYQdQsR1WDGyw7vvvnstmWElpPU380iIlfvEVD1lOwQ6Qh8LoxixmUXqD71vYiaJIKQQdgRx4bqjy2aKb/msSyUyXSPLTyZaSY+Cca76aDPSRHS//farU/9YkTATssMjIYWwA4gL2LQ4wX0zG5TGwI033phCOMPJCuQmt4mg896kzyJppsDcK8w0C7AXKYQdAUtQrMeUKM8YCZhDnELYTTrn4sPcXtcEmmcuwaKLTRTWz3QRhBTCGQyWoBggS1CzAPNLZS5ZgwTQXGIXu7u+4HjbYEnOXDrn6kWbMOVOkbcHQsFMdoebSCGcwdBSS2KE0Jm2xQKIYHmTbYMkOfNJCD2EK8Az0AvRM1i6hhTCGQh3ceTSEEKdSVzk4QI32TZAkt0gIYxHbUqOmF/NJQ5XuAsucSCFcAYimqsq9FXYSwDbBkKy25RRVhMoTKKFlwarXRK/JlIIZxDCEoRzzjmnNvv0+El3+7aBkOw2lddoy2UKoA7T0Wa/i0ghnEEIEZQQ0UlGU05NA9IFTgabIRF9BT1vWZ9CZVRdRgrhDIMpUWYCKKDl5hDCQWunn5wcSoyxAoVK1AqKHXsgfRRMdxkphDMIMoD64Wm9fuuttw5njXUsbhsYye7QzdC1wPIzT5gAbrPNNsO1gl1HCuEMgXpBz6nYaqut6oUOxNDFnxZh0s2QKwxnnXVWrRX0UKrE/5FCOAMgNmj6lPqvuXPnliuuuKIGwc0a4ApljDDpZqhUxgPrPWvEk/ji2kmkEM4IaBIqO3zllVfW+I+LPpgzRpIoDqj/pCayxxxzTL1umlUGXUcK4YBDu3cXNzdnOnvfJfubvIMDDjig7LHHHjWJFkz8HymEAwzuDRHUGZkrnEmRJPICwiOQKVZHKjzivbZrBDCK7hP/RwrhgMJzRtZee+1y5pln1r/NIXbhtw2MZHdI8AifG+NVV11VQyX6EWqm0OWC6YUhhXCAEK6Mp855xOThhx9e7+xqByVLMinSPTrnvVRGFVbfDjvsUGeNNJsrJO6KFMIBQYigO7xnjRxyyCH1b73kCKMB0DZQkjOTcb6jg1BQlYBni4CYoOcqX3311fXvxMhIIRwAhAjqI3jooYeWPffcszZa5eqYOZKZ4W4xLD9lUmJ++k0G/Q1KqbTU0n4NMju8YKQQ9jlCBN3VxXpc2GJA7v4xINoGS3Jm081PSKQNp59+elljjTXmE8G4jhLtSCHsY8TFK9az7bbb1ilRYoHZVivpBtgmhNquKZjWkTwxeqQQ9imi2NWDtHfdddfaUstEedZgusLJNiH0ebNgOjF6pBD2Ofbee++y3nrr1VigOcQxCJqDItk99gqhcpkNNtig7LXXXkOfJMaCFMI+AyswmibMmjWrBrw9ihEkRyI22DY4kt2ha2DevHn1uiCInlW94447Vg8CMiY4NqQQ9hmi4n/27NllrbXWqrNGAimEyaBrQBMFlQTix1tvvfWwCCbGjhTCPkHEBMGskTXXXLNm/5roRyEUr0xOPdULmjmiYHrTTTet9aSJRUcKYZ+BuGy88cb1Idu9YC2aPK8Vvz6D003bwT3z8J+ZSn0ePeBIZx/9/NTu9QNty2677VY22WSTFMEJQAphn0CnYE0UVl999bLRRhvV5pmnnnpqOeWUU8ppp51WrUPu8kEHHVTrCb1OJ20D6n5j+2Yy58yZUx943i+0PbvssktZZ5116vzyxPiRQtgncFc/8cQTy1FHHVUH38knn1z/7uUJJ5xQqU5suhnb0radM4l6PfYTieGxxx5bzj///KGrJzFepBAmEonOI4UwkUh0HimEiUSi80ghTCQSnUcKYSKR6DxSCBOJROeRQphIJDqPFMJEItF5pBAmEonOI4UwkUh0HimEiUSi80ghTCQSnUcKYSKR6DhK+R/fKjhOvRAf+gAAAABJRU5ErkJggg==)
physics-General
physics-
A cylinder of mass m and radius R is rolling without slipping on a horizontal surface with angular velocity
. The velocity of centre of mass of cylinder is
. The cylinder comes across a step of height
.(Assume required friction is present at edge of the step.) angular velocity of cylinder just after the collision is (Assume cylinder remains in contact and no slipping occurs at the edge of the step)
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)
A cylinder of mass m and radius R is rolling without slipping on a horizontal surface with angular velocity
. The velocity of centre of mass of cylinder is
. The cylinder comes across a step of height
.(Assume required friction is present at edge of the step.) angular velocity of cylinder just after the collision is (Assume cylinder remains in contact and no slipping occurs at the edge of the step)
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NRkZHq9+ftJeWRgm0SS8QZVbLqwdJwDBm/lsTSFDyEZeu8oflX1nkzp1yuXDl8/fXXqFSpEqpXr67+LV++PMqWLYsyZcokuvi677//Hjlz5sQXX3yR5Pczxfruu++QK1culcIw5+dNaP3888/qa3nnnXfiFPC41kcffYQffvhBHRanLC/g5bQMYwb0x4D+/dShsFp9+6Jf/0EYNmYKVuxwxrUH0m4q2A6xRJy3u9YQiR86dEhFaYl5g7B1nj4vjKYMgZEfUxFM1wwePFh1Ze7YsUMdyDE6fxMOKA4JCYl3MYLkBWfatGl48eJFnK9JicXpS7Vr11YNWnE9b4nFnwMj63Tp0sUp2Fz58+dXos10GN0YeTDM1JChP0PDCcBGhxrIqf8a0mXNhdx5PsKHefIgj1q5kSvry7uH/GU7YOX5V365gmDVxBJx+oVwQLKloXCyXp2HmZy6/0/8/f1VJQSF4Nq1a9quYfAgjoMVeJDLixcPKGmARe+U6dOnK1FP6sfmOcLvv/+uvNjNBe9S6tevr1IZ1gLPLN5///23RDtfvnwq7cWfJVNPu3fvVmkX80MRr4as+q/pxx6zsHXfATjt24u9+/Zh797d2LxiBnrXLoZM+uc/KD8cZ9+aTiQI1kksEWcnIqNSa4AiRYMqVjkw4maH4pQpUzBkyJBXlSonT57UXp18WCJ39OhRFdnz+8DzAV7UuOjsSJOsq1evaq9+G14MeCG4ceOGtpPyXL9+HQ0bNlSWANYA70KKFCmiovAaNWqoyhT+vPbpRdIS1TuxCcCmodWRTS/SdvMvIM4E3IOdaPt1dv3F5xMM3CPRuGD9xBJxihX/8KypxpdWAGwvZ4Ts4OCghgSvXr06xf1LGC0eO3ZM3Znw+8L8KS8cLOvjxYWiHhMFh4aGKl+V8+fPq8fmgCkhfj081LQG+D3gz4UmYdbyNb3NaxGvO+sM4u5z9ceC5p/j3TQZ0Whu3BdsQbAmYon4jBkzVEWIOdMChmDJiwujSUbc69atUxcVijqFu27duuoC8+WXX+pvy/eqskNzDJhmWoJ3DJZJTdgir0W83hxXhGi7b3MRY6p9jDS699BprXiiC9ZPLBFnJMWDPuachfihSLN7lEOLWerIKhmmElgxwhQHH9ODnG6Q/F4ySjU1rJHnnYk1OCraBq9FvI7jCQSEhiFM/3PhzyY0NATBgQFwWdwN/8uoQ9qPG2Kdj+l/ZoJgamKJOBtqmH+2ztth64V5X+bQ2aDEqTs8LKXbYYUKFZRZFlNAjOCZAmEKhvnj5MJqGJZXpsQFInVCEa+JXHoR/7BEZdRr2Fh/wW2gLrp29Wripy//i4z659JmKYI2c09BCg0FWyCWiLu6uirx4aGZkHToF8K8+ZuCytQPx7tx+o6jo6M6jOWBH8sR2XHKzkVeNHlYakyjC6NwNtaYI3WTOtCL+LBaeE8v1Jk/yI9PCxVGoYIFUZBL//ZnX5ZAmZr2+GO1Cx7Lt1SwEWKJOPOrv/76K9zd3bUdISnwoJORcWKCSttYHpby8Jhdnqy8oa0AU1jcY5NTUgdy8KLA9JeQVBiJ10B2vYhXGLoGpzy84XHBXf2uX/C8iBt+z6VrU7A5Yok4oYgzQhSSBg882bW5atUqbSfp0F6AjU28ADCN9WZZI2uqN2/erJp6/gkjdx6o8n2FpPI6J15/njvCtV1BsGXiFPF+/fqprk25TU8c5rZ79+6tmlpYLZJc2Ll45MgR9f3n4WivXr1UxM3OUqZPaP5FvxZO62daRtwBDeG1iMdfYigItkWcIk4BYfehKQ7fUjP8/vD7lDFjRmTPnl2VH5oaeoXz465fv16lWyjq7NKkXwq7IOOK0oX4EBEXUh9xijiNoTgsmbf6QtxQwHmQGePW980336R4tya9RVjWSK8UNvkwn/5PnxchIfyxbuAveEf/86o2zQUyu0pIDcQp4rR+ZWmcRHlxQ6MnNve8abdKQ6fYh8HRiAwPRdDzZ3geGKgONd9aQcEICTPOn4O+7zSaEgzhKZymd8L3xYqhxypPKSEUUgVxijhz4fb29qpSQngbmmYxNx2XXzZtWGnjy9b7sLCX4+0uOS1E35ZN0bRJIzSws4NdzKKDYpPm6NBzCKYu3opzt5LesHPnzh1V2cImI0EQ/t3EKeKEfiEDBw7UHgmEAk7zrYQGHnBx6EGnTt1x594lHJ/dGrn1e2mzf4RPi3yGokUKo3DhIqq7s1CBfMie9uX7/Pfn7th2NUj7TAmzceNGFYnzgFMQhH838Yp4zDR3mXr/EqY/KOAZMmSIJdpvrty5c6vSv+EjxuDmLXecmNceH+r3i3eZh8NuXlpd8gXlW+527iT2rJuNdmU/Ue/7dZfVeJSE7AoPU3k3IAiCEK+Is9OQbn20Zv23k1gEXqBAAXTr1k3N2WQjD793L4nG4RmtkEf/mp/HHI63GiLo8CgUz5EW6QrYY++zhMs66cPCKFyafARBIPGKOJ346NTHWmWbIjoS4eERiFJaGI1I/f8jMhnl7jSX4mT9GMFOmzatisbpWMhomFNpWKsdt+tjCI7MtFciXm6EE+K1FLu+GJXzpse7ue2wMZF+b9rysjLFnL7lqRWe/fD3XJYsa1ociGOIW2u8Ik4YhdPrw1ptad8iKhj3Lp3E5mXzMG/NIfiq9PIznNu+FLPmLIWThx/CDBRzNt5UrFgR7777LrJmzYpSpUph/PjxyuSKXZr8hifMaxEvO2IfHv3z80dHISw4APvH1Vd+HrmrjMelBCxUKDq8sDLql0as5MMDYlYZsVmLDW6yZFnD4p32mjVrkizkCYo4fVQ6duyoPqA1E/XsCtaNbo1KFe0wdMkR+D1/Dn//xwgMDUdY4C1sG1MPBQtVwOidN5GY7L4JR7MVYzlajx5q1qfhFzOKeGvk0wt0gaodMXyiI6ZMmoiJk6ZgmqMjJo//HR1rlVDNJ5nylsXYvbcS9O7gNHlG4Ry6ICQfX19fVaa5du1a5TBp6GIDFldcz8mSZexauXKlChSTSoIiTti9ycYfSw5jSJDnXpjV/jt8WLgGph28rbYCTsxHK7uu+Pvcy+qN4KvrUTtHGhRpOgWeZjXM0Iv4rLb4WC/SadNnQuYs2ZAtaxZkfDcmn54W7xUqgSrNemOe09W4x4W9AUs+OYTCGMdDITb0e5cKLMHWSVTEOfSAnYH0yLY+ArB7eHXkzJgX9os9X0axz9wxvl5+6DJUwGLXAPWqkFtOaJVXh1zl+2G/WWddUMRbI69esIvYOWDeyrVYu2YVls6bgj71S+L9NGlQtNogHLqvvTwB2CHav39/NXlJMA08HKaNgTHQcphREyc5CYIlSVTEGfUxb8jqDGvj2ZnpKP/hO8hRZhguaLO2Ao7MxC9Zdchdbxo8tUqPR+5zUDq9DgXqjIJL3DO5UojXOfHyY4++3SEYeAmz25RAJt07+LzeeJzzT/gWgSWJVapUUV4qgmlgmpD5cGPgYI/MmTOrgRKCYEkSFXGyf/9+VW7IvLD18ARbBlRAFl0m1J/n8SqX7LGyF4rm+B8Gbr6Gl0mHMLjNb4CMuuyo7rDVzKZHbxxsDt+Lh/+oAQ/13YHOpd6DTpcVlX7ficfxJOzpmcIhEvQsF0xHckScLpJMidGMjNUEgmApkiTijMY5kJcDlK2GF6cwuHJ+pMlUDosux7Ssh8N54SDYd9VH4TERd/AljC2fFbq8lTDz1MtYODoyHGGh4QkeIpqGhEUciMDVtf3wZTYddJm/x4QjcedVWMLIKhmZtmRakiPi7AegiBcvXlwJuiBYiiSJOKHHdYMGDdRtvVXweA86/5ADus964bzWVBr1yBmjOrfBsGWvjah8NvdGgSwfoObv25VrXdDFA1g82xGOU6fgr73eCE5RJU9MxPVE3MTCdt8qZ72Pqo6Eaxy3Ciw7Gjt2rPZIMBXJEXGO26OIZ8qUyeqrt4TUTZJFnLDUjmJiFTXK0d6Y3Kgo0n3SBFsu+MLnhjv2rJ2KDs27YOTM9bjq64srzsvRoUJxVOoyF960Rg85g0m9u2Lcalc8fuKM31t2wtLzKWm3+wIHJjdABv0f+zcDd+BBPC31oZ4rUO9TVqukQ+URexHwxt05S42qVq0qUXgKYKyI85CZ069eVhjpMG7cOO0ZQTA/Bom4m5sbatWqpYYpWwO+h+ahw6+10G7wWEyZPgfrjt3Ao+vOmDeqHwYOH4X+XTpiyMwduK1V5D10mog6tZphhRuTz4FY3KkM7MYfRRLsSowkDB7bp6Jl7doY9LcrnsZ77QvH2aWDUb9WDTTuOQNntewQyzo5vYej26y2xNOGMVbE6emeN2/eVyLOkXpS9ilYCoNEnPCAjR1Fb051tyQRj33g4e6Fe4GvFTLs8S1cOOcKn6dvCl80XJd0RYUqHbHdh49DsWFARZTuuDjR+mxLwSYU1oWzKUUwPcaKuIuLC3LkyPFKxL/99ls1qEMQLIHBIs5JMnXq1LHBPGA0zizsgIpVu2H3HT5+gfUDKuCHdgutUsQ5kIO37OwIFFIGY0WcY/LedLOkMdr27du1ZwXBvBgs4oTDCJinZa2sLXFzy++oVac1Nlzho0As7lwa1YfuSsF0inHwzGHw4MHo06ePmiIkpAzGinjt2rVfCXjMmjNnjvasIJgXo0ScsAGIB522RITPDvRt0xZTdt1GeNgFjG7cABMOWV/zDKM6nj1YV11+6sMYEaevPAd6/FPEmWLkc6YiKuQJbl/1hLuHB7y8PODhfgEeXpfg4/cYIUk1AIoKweVdszBj+1VtIxECr2HP38uw0+0+pPLddjBaxGnGxOEH9NC2HcLhuWUu/pg0C0sWTMK4eU7ws47U/iso3Py+Llu2TNsRUgpjRJxDxPPnzx9LxD///HNcuaJu8UxCdPBDnF49FFW/zIvcHxRFnW4O+H1gV9QrXwo/1uqFv13uJmqxHBl0FXM7NcH4I4+0nYQIwfGZbVAk8wewX+Iu80dtCKNFnBw8eFC1gtvW4IhIPLl7HVdu+iHIEEtDM8ApSnSNZGemWM2mPMaI+IQJE5AxY8ZYIs51/Phx7VWmIBpRwQfQlR29matixZUnCA4KhM+h2aid711kLNkdu64mnGp77rkY7VqOh+vzxE59ohBwahmafpEJaXTZ0WqJG1iRK9gGyRJxwlmcPOi0tfy4NcIa/BYtWshIPDNhjIjzLomC/f777+P777/HV199pbzmubd8+XLtVSbi6krYFcuM3LVmwDfmjjHiIma3KqT/fAXx286LCZznRMJ1Rnu0dTyG4EQOfcIeHMOUgb3RtOYvKJorO5r/JZG4LZFsEadvBA/h2rVrp6bgCMbBcsJq1apJU48ZMVTEHz9+jI8//hhff/01/v77b1W/P3HiRCXe2bNnV/umHF59Y2VvfJ4lPWpOP4dXMXegOybZ5dGLeDGM3JOAfXG4Fya0bo+FZ+4nbC8R9hA7JnZB9yl74bR8CIpnzoAmi0TEbYlkizhh8wM9x5kGsJb6cVuCaSl6o9BoTDAfhoo4h6TQCtjd/aWtwx9//KGGShCWgvKgnxOfTEMAVvYuh2zpS2HG+ZeWyuTe3vH4LpsO75X7DcfuxZ9OeXFmKlp2nYULjxI6oozAzX1T0bbtUBx/BPiu6YXPMqYXEbcxTCLihHXNdHTjL7Z0FyYdetLwXIGRnWBejEmnxMDWe6a/pk+fru2YGP+D6PvLe9AVaY8dF+7g0W1P7PxrGKr87318+F0rLNNH2PHLcziOjG2P3ouO4kkCf4ov7hzEyLZtMPnAy7sH72U9RMRtEJOJOLl69arKj0+dOlXbERLi5MmTKoWyYMECbUcwJ9Ys4gFHpqPSRzrkq9IJw0f2RvUvC6PINzUwZOE2uPsmkrZ8cQLD2vXBipO+8adSwh9g27hO6D3rmGbZDFxZ2Quf60W82VJvg8YYCpbFpCJOPDw8VDPEzJkzJSJPgFOnTqlacN6O0y9cMD/WK+IhODqjOT5Ikx+915+H34PzGFgiAzIUrou1d7WXJMDT/ePQbuBsnLkf3+9VBK5sHYmmbcbgsM9TBD17iqdPn+LU3E4oqhdxuxnH4fcsCKGJ1TAKVoHJRZzQIItT8jkZPihIjZ0X3uDQoUMqAk+xW3EhSVitiId6YWaLT6H7qBE2X3mZNDkzsTLSZy2K7iuuqcfxE4itI3rAYcFexD+J0A3Dfv4YufJ+jh/LlcVPpUujTLlyKFEkLzK/kxa5CnyBsrW6Yc0l6Ra2BVJExAlnc9J/nK3jPPgUXsIqlMqVK2Pp0qXajmAprFXEw7w3oVUhHT6wc4SH1gQa4DodJTNkQqkOC5Fgj3HAPjj0HIjFhxI6YL2KlXqhb25nh0aNG6upXY2bNEH174sgm17E85WohMbtf8f26+LMaAukmIiTO3fuKCvVVq1a4caNG9ruvxemmBiB79q1S9sRLIl1ingkLm4ejMJpMqPelIOIyX5HPvXAqJ/+g/98Y4/NCZha3t02ET2HTMGJ1wUtSeb2un4oljE9Wq6Sv1VbIkVFnDDXxiHLPPBkGuHfiL+/PwYOHKgaRU6fPq3tCpbGGBGPOb9gKe0/RZypw4iIZB4JRt7Fyh7FoNN9hdG7bmqbeqKe4eiESkjz7sfouNA9niaf+1g9dgCGztinplgZitfS7iia/l001n98qU6xHVJcxAkbghYtWqScD9kg8W8y0OcsRnqCd+/eXe5GrAxDRfzevXto3bo1Vq9ercT6zz//xNy5c9VzDg4OKFOmTDI6l6Pw+MpRLB3XCSWys43/fVTp/Cf2XfLXKkWi4XdqJkpn0uE/BStgwLRN8HgQ9Fb1SdTN7RjetzfmHjWu49dzSVcUSqtDgwVuIuI2hFlEPAZGoUyttGnTBhcvXtR2UyfBwcHqgsUacF7ATOlwJ5gGQ0WcKZSiRYuq7swBAwaoSJzdyvwYadKkUWdAxlsmRCPs2QNc83bD6VNnce7cWbh5Xsa9p6Gvou6osOe45XEWp06ehpv3LTz5h51hdJA/7vjewRMjY6SQgNvwOu+KG/4vEmjnF6wNs4o4YesyG4IobmxXZhdcaoJ3HSwfpA1B06ZN1YxMwToxJp3CzmT6pKRLl06NaMudO7cScO5t3LhRe5UgmA+zizihQ9/hw4fRsGFDNG/eXB30Mcqxdby8vDB8+HBVfcL2bDGysm6MEXE2ZmXJkkWJ9puLoi7nHYIlsIiIx8Dc+F9//aXGkPXs2VO1oNsiNK1i6oT/j/79++PatcRqeQVrwBgR54X6k08+iSXiP/30kzh5ChbBoiIeA02DRo4ciUaNGqkDwM2bN9tEtyeNkJgXbdasGTp06CAGVjaGMSLOdNkXX3wRS8SZGxfzN8ESWIWIx0BR5Ik/84701aZXuTUeCLLipFevXqpJom/fvtiyZYsMcbBBjBFxwt/Nf4q4bU24ElITViXiMTA9wY5Gzi2sWbOm+kPbs2ePRVv4eRs9efJkVS7YpEkTjBs3Ds7Oztqzgi1irIgvXrz4rbx4tmzZ4OTkpD0rCObFKkU8BlauME/OTkfW5/LAcNCgQdi5c6eyvn3y5InJb2FZ/8von+b+Li4u6oDSzs5OmXqNGDFCRd0ywDh1YKyIe3t7I0eOHK9E/JdffoGPj4/2rCCYF6sW8TehoLu5uWHOnDmwt7dH+fLllX85mywYtTPFQQdF2uH6+voqvxZWh7DqJSQkRIk9F99mRM9SRzZv8GLAP0qWAlKgOUOR5YH8w+RBJatN9u3bpywE+L5C6sFYEefvzpsT7/kxJJ0mWAqbEfG4oGCzNpcRMmuy2WxBAWaemnv0NafV67x581RpGP9lhx1bpVmrzsOorl27qoNJRtuczMLXsc5bygNTP9u2bVO/A4bCYIB3hjEiPmXKFO0ZQTA/ulWrVsFWFtudGT2tW7dOLY7E4tqwYYP6l7lK2t9SxHn4xPRL6dKl1UDbkiVLolSpUvjhhx/w888/qyieAk6xZ3TPjxvzcWIWPwddB/l54/p6ZNn2YllrpUqVVM+CofDujwKeM2dOld4TBEuh69y5M2x98QCUi6LMEkX+cfIWl2LO1a9fv7cW92iRywoTRt/dunVT7xvzceL6HLJS32JKjoOPY+ZkGsLZs2eViDNIYKpNECyFjq5ssmT9mxbz11xHjx5VKTZjYCovbdq0qsxU8uGCJbHpnLggJAc27hjbh8BD8YIFC6pDdkGwJCLigmAE7ChmqalYLAiWRkRcEATBhhERFwRBsGFExAVBEGwYEXFBEAQbRkRcEATBhhERFwRBsGFExAVBEGwYEXFBEAQbRkRcEATBhhERFwRBsGFExAVBEGwYEXFBEAQbRkRcEATBhhERFwRBsFmA/wMgHoDd7oXCjwAAAABJRU5ErkJggg==)
physics-General