Physics-
General
Easy
Question
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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)
- an ellipse
- a circle
- a parabola
- a straight line
The correct answer is: a circle
Related Questions to study
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,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)
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,iVBORw0KGgoAAAANSUhEUgAAAL4AAACkCAYAAAApOZRgAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAFxEAABcRAcom8z8AABH1SURBVHhe7Z0JeE3X2sdT89QaI6ihQSmKa6ihRJoqt4Zya/pMMTRoiaEexS0aNRMS1BBa0gb5IqYGQWJqXEqCmmtI4hpCgxgTIvP/y7uspHxOyLDP2Xuf/f6eZz15zrvWWWftfX5nZ++1117LBgxjQFh8xpCw+IwhYfEZQ8LiM4aExWcMCYvPGBIWnzEkLD5jSFh8xpCw+IwhYfF1xO3bt7Fw4UJcunRJRvJGeHg4Fi1ahOjoaBkxDiy+jvDx8YGNjQ3GjRsnI3lj8uTJor6VK1fKiHFg8SXXr1/HhAkThAyLFy8WR9a8JE9PT0ybNg0zZ84UR1VTZXKSqI5OnToJURs3bqxInc2bNxf1tWvXTpFtzm2ifTV//nxs3boVKSkp8hsxLyy+ZMmSJUICTuqlokWLIiwsTH4j5oXFl9BRh3Z+1apVMXToUAwcODDXaciQIejQoYOo780330SvXr3wxRdfmCyb3UTvb9q0qaizZs2a4vWgQYNMln1dovcNHjwYtWvXFvU1atRItNlUWXMnFxcX9OjRA8WLFxdt2bFjh/xGzAuLL1m2bJnY8b1795aRvEFHrnz58qFWrVq4c+eOjOaNBQsWiDaStEowYsQIUd+MGTNkRB2ePHmCli1birYEBQXJqHlh8SVLly4VO75Pnz4ykjeeFz8mJkZG84aHh4dZxKfrEDVJSEhA69atWXw1yBBfqSP+kSNHMsW/deuWjOYNugCkNtKpihIMHz5c1Dd9+nQZUQc64rdq1YrFVwMWXz1YfBVh8dWDxVcRpcU/fPiwqI96YJS6M/rDDz8oKr6WLm5ZfJXw9vYWO37AgAEykjciIyNRokQJNGnSRLGbMv379xdt/PLLL2Ukb2jl4pZwcHBg8S0J9Sj8/vvvmf/2P/roI+zevTvPadu2beLO6Oeffy6+TFNlcpLoNCd//vyijXSPwFSZnKYPP/xQ1NelSxeT+ZZIwcHB2LNnT+Y9ChbfQqxfv17scHOkggUL4o033jCZx+nFRPuK/rL4FuLo0aP49NNPYW9vL3a8ra0tnJycRL9ybhP912jQoIGor3DhwuLI7+joaLJsdhK1p06dOqK+kiVLivPhNm3amCybnUSnFfT+ihUrijqrVasm2myqrLkTtaVt27aws7Nj8dUg4+ZQz549kZycjKdPn+Y6JSUlISQkRPTqvPvuu7h27RoSExNNls1Oovb4+fmJ9pGwDx48EKdopspmN9H7hw0bJup0c3MTbTZVztyJ2hEXFycODCy+Cnh5eYkd37dvXxnJG8eOHRPi03iYe/fuyWjuCQgIEO2jo79SF8uurq6izlmzZsmIOtBBgXt1VELr/fibN28W7aNTEjpSKgH34zMsvoqw+CrC4qsHi68iLL56sPgqorT4dFOM+vCVGrLA4isLiy/JEN/Z2VlG8gYNWaA+fOrPp+7IvLJr1y7Rvk8++URG8s6YMWNEne7u7jKiHvSDZvFVIGMAGPWT084n0fKStmzZgmbNmqFr167YuXOnyTLZTdSeKVOmiPbRD4keyjZVLieJ6uzcubOok8YnKbHNuUn0uf7+/qhRo4ZoC8UsAYsvyTjiK5noiF+gQAGTeZxMJ/ohWAIWX3Lx4kUxWKtFixaaS3Rrny6SSYy33npL/CehZ1RNlc1JokFqVDf9NZVvyUTbQ49UWmpyKxb/OeiO6KNHj8QtdC0luvhbt26dEJ/GttAzvKbK6TXFxsaK/W5JWHydQOf1JP7HH3+M1NRUGWVyC4uvE8zRnWlkWHydwOIrC4uvE1h8ZWHxdQKLrywsvk5g8ZWFxdcJLL6ysPg6gcVXFhZfJ7D4ysLi6wQWX1lYfJ3A4isLi68TWHxlYfF1AouvLCy+TmDxlYXF1wksvrKw+DqBxVcWFl8nsPjKwuLrBBZfWVh8ncDiKwuLrxNYfGVh8XUCi68sLH6WpOHRnSicCj2AXcEhuPlYhlWCxVcWFj8r0p4iPGw7+rWsgnKNB+JSgoyrBIuvLCx+lqQBSX9hpIMdPhi+RsbUg8VXFhb/FcRfCUQz2wqYuiNSRtSDxVcWFv8VnPH5CqXeaYtDfyXJiHqw+MrC4mdJIjx6/QN1/zUZ92VETVh8ZWHxs+J+KNrXrgRn970yoC4svrKw+FkQFeyJ6nbVsPK4yv2YEhZfWVh8k6Rh43edYVenDyLyvpiJIrD4ysLimyLlFka2skXzketkQH1YfGXRvPg0N3xoaCj27duHU6dOyah5iY/YiPplKmPW9vOIe3AfCWkyQ0VYfGXRtPi0Sknbtm3FF06Jls8cMWIEHj58KEuYhxNeg5CvVB3MWPwLtu07hlgW3+rQrPiPHz8Wy8NkSP98+vrrr2Up83DKdzLqNWiK0Z5b8ED9LnxBQECA2HaT4qfE4cKx/fD1WQ3vn3/Gjyt/hLfPWqxd8wt+WbseIccuQOURF5pDVfGT458gIcX06h4ZXzQlWi+WjvYZrytVqoRr167JkmYg+TGio2OQIl+qDS1RtGnTJrHtjo6OL4ufloCbESfh6fqJKPOPbmOxftOv2Oy/FvO/HYL677wNh37f4fQdjfyKNYCK4ifBx20IvlsTJl+/yE8//ZQpev78+VG8eHHxA8iI0ZI4PXv2RLdu3RRP3Xv2Qt++fdCzR3d0N5FvidS9e3exffTXyckJDRs2FNtNp36JiYlyL73I9aB5KJZeZrDXIRl5xplNk1EkPd58mBeeaOC0TQuoJn7SX/9BqzI2KO80Abdl7HlOnDiBYsWKZYrO6VmilQHT0kzbezlwLkqkl+nn8f+WzEy6jP+xT3+/3T8RFi9jBkc18Q+uHovyRQrBplh1rA6NkdEXcXNzM/nld+jQQfxHWLFihdWllStXYs2aNRg0aFDmD79Ro0Zi9XFvb29cvXpV7p2Xidw+B8XTyw/8Yb+MPCP1xm40LW6DwvUHIYLPdgTqiP/oDEYNGASPhd+jej4btHBdBdP/vCG+7Hbt2om1UKtWrSpEmDdvnsy1Tg4cOAB7e3uxrbSE/82bN2XOq4ncPhcl6cAwbgX+PH8BFy+cx9mwYHzzeWNUrNoEC7adkSUZVcQ/7fcdun+1GPFpDzHe0RY2pVvit5uvv0U6atQoIcO0adNkxPrw9fUVizjTdnbt2hV37tyROa8ncrs7bN+wQbVmn2HC1O8xaZwr2jepJur6oM/3iHjIJ/gZWF78lGi4DfgXPIKe/cs+tW4sCqR/MX09fxOvX4W1i798+XIUKFBAbOPAgQMRH5+zE3I61aFz/D7zApGcmoKkhHjE/HUVWxaOQumC+WDfxgXHbvG5DmFx8aP/sxztO7ngj1txYjXre5F74VjeBiXq98elOFkoC0aOHGm14s+ePVtsGyXazoSEnPe8Z4jvvHCPjGTwBO69aom6v1p+UMaMjYXFT4D3mI5o2KYbJk6aiG+++QbfTp6Az5pWTP9SymLm9nBZzjTWKD710U+cODFT+ilTpsicnJPZq+MZLCN/s/HbZ3fA+83bKSPGxqLix0dsQ8/P+iI4PAYP799FTEwM7j+IRdSR1ahRyAb2XWbgkSxrCmsTn/rjhw8fnik99dzkhaigeSiaXs+ApQdk5BkPwoPRrlL6Z5RsiA2nH8iosbGo+FtnD8WQOVvlq+dJxMK+9WBTsBp8TsXK2MtYk/g03qhfv35ie6jbctWqVTInF6Q8xsXjIZjq/KGoz96hFxakXy94eXlh7pTRcKz/Dmo0aA3PX/+Qb2AsIn5KQizO/+aFumXt4PrTftyLffGW+5OHMfCZ8Ox2e6WPR+BwRBTiTYwXsBbxb9++jc6dO4ttoR6cDRs2yJxckhqPy2dCEbg1AIE7duDXTf5Yu3Yt1q1bB1+/Ddi57zCu3cuqw9iYWET8pzER8HH/N0a4joKb+0qEhb94r/bG2QNYPH0ixowZnV5mJDx9AnHLRIeGNYh/5coVODg4iO2oWLEigoNfPh9nzI+FL27zht7FP3fuHBo3biy2gW5QHTr04pgaxnKw+Bbi6NGjqFXrWZdi/fr1ceYM30VVExbfAuzdu1cMpaa209CLiIgImcOoBYtvZrZu3Ypy5cqJdtO4mxs3bsgcRk1YfDPi4+ODokWLijbT2Pr797UwNRVDsPhmYsmSJShUqJBor4uLi3iUktEOLL4ZmD59umgnpXHjxolhCYy2YPEVJDU1VYw/ypCe2pnV01KMurD4CkGnMnRKQ+2jZ4QXL14scxgtwuIrQGxsrLh4pbYVLlxYPDXGaBsNip+GNNMzjmhSfHossH379qJdZcqUERM/MdpHG+KnJeHy8d2YPdYZ3Yd9j/B7ps3Xmvj04Hfr1q1Fm3jcjb7QiPjJiIkKQ5/3y6D6P/+NGB0c8WkeTxp6QO2pXbs2wsJMzw/EaBPtnOo8OonPGlbGAI8XH6J4Hq2If+TIEdSsWVO0hSZ6Onv2rMxh9IJmxL+2ZxFqlK8B7zNZP2CtBfGDgoJgZ2cn2kHT+Zl1KkPGbGhG/E1TO8K2/mBcfcUsI2qLv3HjRpQuXVq0oWPHjoiOjpY5jN7QhviptzCyRXm0Hve/MmAaNcWnmduKFCkiPr93796Ii3vNlBCMptGE+IkR/qhTtgpmBWpzlgUPD4/MCWuHDRuW4/luGO2hCfGPLXdB2ZqOOHAzi+4ciaXFp+EGU6dOFZ9Jafz48WJYAqN/NCB+LGb0aIz3O0167XqylhQ/OTlZLECRIf2sWbNkDmMNqC5+6u1QtH+vDPovCpWRrLGU+DTuhmYrps+iocXLli2TOYy1oLr4kcHusC/7Hn4MOYbDR/9EvMq9Onfv3hULM9Dn0GIUNEUHY32oLv4+DxrcVQ79v12AQ+eikPqKUbzmFp/G3dDjgfQZtra22L59u8xhrA3VxY8+HYy5c+Yj6NTrn0U1p/i0wmKzZs1E/TQPf0hIiMxhrBENXNxmH3OJf/LkSdSrV0/UTeNujh8/LnMYa8Xw4tPqI1WqVBH1NmnSRBz5GevH0OLv2rULFSpUEHXSuJtXrS/FWBeGFd/Pz0/02lB9tOQOTVnOGAdDik8rC2aMu3F2dhZTdjPGwnDi04qJVAclWlMrKYnXhDIihhJ/0qRJmdLTkjs87sa4GEJ8Gk2ZseQOjbKcP3++zGGMitWLT+Nu+vfvL95H893Q8jgMY9XiP7/kTqlSpURPDsMQVit+VFQUnJycRHkadxMYGChzGMZKxT9//nzmkjs1atTAwYO8qDHzIlYnPo2zqVOnjihXt25dnDhxQuYwzN9Ylfj79+9H5cqVRZmWLVvi8uXLModhXsRqxA8ICOAld5hsYxXir1mzJnPcDT09xUvuMK9D9+LTkjvUP0/xwYMH85I7TLbQpfhz5swRr2nmA3pNiWZE4HE3THbRlfiurq5Ccjc3NyF/hvT0mmFygq7EHz16tBA9Y7FkOsVZuHChzGWY7KMr8X19fTOn8qPx9LzkDpNbdCM+zXfTpUsXIT3NWLxlyxaZwzA5Rzfi+/v7C+nffvtt7NmzR0YZJnfoRnya7InG0fPUH4wS6Oocn2GUQjfi37kUCu+l88Vy+XPcPbHKbweu3L6PyNOhiLjLjxAyOUP74qc9xoY5w/FBUweMnbkUwft/w4GQEPy6djlGdG+FKu+1xpYLvGw+kzM0Lv5TbJraHfkLVsDcwEsy9je3TvjBsWkLrD78QEYYJntoWvxbR1ejio0NGn2xAlmdzOz3XwrfvTz8mMkZGhY/BT+PaAEbm0KYuu2/MvYyT+Mf4t49HpjG5Aztip92FcM+oKU1q2Ptn3wqwyiLdsVPPoVetYqmi/8+Nv83VgYZRhk0fKpzHcObl0kXvzJWneQHSxhl0bD4qfAf31YMUxi68piMmSYx8Sm4Q5PJCRoWH3h0cRualLVBiXp98GcWExqf2r8N+/64kmWvD8OYQtPiE3/4T0WVIvlQvd1w7DoRjkdiWcRU3IsKxw7fZXD38sfNx69YKpFhTKB58YlLB/3h0u0j1GvQFB269oLLl64YN2ESlq7dheinshDD5ABdiP+MRNyOuoLz587i7LkLuBETJ+MMk3N0JD7DKAeLzxgSFp8xJCw+Y0hYfMaQsPiMIWHxGUPC4jOGhMVnDAmLzxgSFp8xJCw+Y0hYfMaQsPiMIWHxGUPC4jOGhMVnDAjwf+Bouv4j0xllAAAAAElFTkSuQmCC)
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,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)
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,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)
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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)
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,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)
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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)
physics-General
physics-
A Cylinder of radius R is spinned and then placed on an incline having coefficient of friction
is the angle of incline). The cylinder continues to spin without falling for time
![](data:image/png;base64,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)
A Cylinder of radius R is spinned and then placed on an incline having coefficient of friction
is the angle of incline). The cylinder continues to spin without falling for time
![](data:image/png;base64,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)
physics-General
physics-
Portion AB of the wedge shown in fig is rough and BC is smooth. A solid cylinder rolls without slipping from A to B If AB=BC, then ratio of translational kinetic energy to rotational kinetic energy, when the cylinder reaches point C is
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)
Portion AB of the wedge shown in fig is rough and BC is smooth. A solid cylinder rolls without slipping from A to B If AB=BC, then ratio of translational kinetic energy to rotational kinetic energy, when the cylinder reaches point C is
![](data:image/png;base64,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)
physics-General
physics-
Four massless springs whose force constants are 2k,2k, k and 2k respectively are attached to a mass M kept on a frictionless plane (as shown in figure). If the mass M is displaced in the horizontal direction, then the frequency of the system
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)
Four massless springs whose force constants are 2k,2k, k and 2k respectively are attached to a mass M kept on a frictionless plane (as shown in figure). If the mass M is displaced in the horizontal direction, then the frequency of the system
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXYAAACoCAYAAAD0HcWOAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAFxEAABcRAcom8z8AAD/VSURBVHhe7Z0HmBRF88b5DH/MOaD4KUHFhCKoiICYUBRRzAoqiuAHImZEJChKFBQDRsAAigQRwYgiyClIEAWRJDlLFI5whDvqf7++rWUY5o5Tdrmdod7n6eduZ2ZnZme6365+q7q6kBgMBoMhUjBiNxgMhojBiN1gMBgiBiN2g8FgiBiM2A0GgyFiMGI3GAyGiMGI3WAwGCIGI3aDwWCIGIzYDQaDIWIwYjcYDIaIwYjdYDAYIgYjdoPBYIgYjNgNBoMhYjBiNxgMhojBiN1gMBgiBiN2g8FgiBiM2A0GgyFiMGI3GAyGiMGI3WAwGCIGI3aDwWCIGIzYDQaDIWIwYjcYDIaIwYjdYDAYIgYjdoPBYIgYjNgNBoMhYjBiNxgMhogh5Yl97ty5MnDgQFm7dm1si8FgMBjyQkoT+9KlS+X666+Xgw8+WBo3bmzkbjAYDPlAShP74sWL5ZprrpFChQq5Ur9+fVmxYkVsr8FgMBiCkPJSzLx58+S2226Lk3vt2rVl0aJFsb0Gg8Fg8CMUztO//vpL6tWrJ3vttZcj95o1a8rMmTNjew0Gg2HXsWrVKunevbtMmjQptiW8CAWxAx76o48+Kvvss48j92rVqsn48eOlV69e7mUMGTIkdqTBYDD8MyxYsEDuvfdexy1ly5YNvSoQGmIH69atk5YtW0rhwoXdCzj77LPdX0qlSpViRxkMBkP+MWXKlO18eTVq1HBEH2aEitjBpk2bpFOnTrL//vvHXwSFF2MwGAz/BCNHjpRy5crFeaRBgwYuGi/sSGliT09Ply5dushnn30W25KDzMxMeeONN+SQQw4xYjcYDPkGo37FV199JSeffLLjj//7v/+TVq1aRSakOmWJfcuWLfLKK684Tf3444+X/v37x/ZswwcffBC33I3YDQZDXqhVq5azzoms+/DDD+XYY4913HHkkUdK165dZfPmzbEjw4+UJfatW7fK559/7h46D/+oo46S999/P7Y3B6tXr5bDDz/ciN1gMOwUJ554ouMKCpMe+VuiRAnp16+f45soIaWlmKysLPnyyy+lWLFi8ZdBz4oUA/7++28jdoPBkC+cccYZcWKnYL0PHz48tjdaCIXz9Pvvv5fTTz/dvQykmRdeeMFJNehlRuwGgyE/UGLfd999pWrVqpGIV88NoSB2MGrUKDn//PPjvW3btm1drOnRRx9txG4wGHYKJXbkl4ULF8a2bgMKQVpamixfvjy2JbwIDbGD3377TapUqeJezn/+8x+577774lqZEbvBYMgLSuxnnXVWbMs2ZGRkyIsvvugUgNtvv12WLVsW2xNOhIrYwdSpU6V69eruBXmLEbvBYMgLSuxnnnlmbEsOmNXetGlTF/LI/nPOOcelCw8zUpbYcZASBfPQQw/J/PnzY1tzwEOnVzViNxgM+UUQsZNk8K677orzyOWXX+5SlYQdKUvskydPjk9AIunXjBkzYntyQGKw+++/X/bee28jdoPBsFMosZcuXdp9JpXAFVdcESd1ssjOmTPH7Qs7UpbYV65cKQ888ED8oQd5sbHklfyN2A0GQ15QYj/33HPl559/3i6VQKNGjSK11kNKa+zr16+X5s2bxx9+hQoVZOzYsbG9OSkHjjjiCLfPiN1gMOQFJfaDDjpIihcv7v7fb7/95JlnnnFcEyWkvPOUab7t2rVzsae8CDI6jhgxwu1bs2aNxbEbDIZ8QYldyzHHHONyTkUplYAi5YkdMN2XvDEHHnigeyElS5Z0M8Y2bNgQTzlgxG4wGPKCEjuh0qeccooMGDAgtmcbmPQYBaIPBbErunXrFrfQGUqxwIZJMQaDIT9QYi9SpIibiOQHaywTkIE0Y3Hsuxkff/yxHHfcce4FHXDAAa73NWI3GAw7gxJ70ASl6dOnOw5hP4UEhGFGShM7Gjq52P3e6sGDB8tJJ50UfwkUI3aDwZAXlNj9E5TGjRsnF154YZxLmNEe9rDHlCV2tK7GjRu7lAF169Z1ceteoLH/73//i88WM2I3GAx5IYjYWStZEwwSIUMUnncxjrAiZYkdx2idOnXivegdd9zhZol5wVTgww47zIjdYDDsFErsZcqUcZ+Rdf/73/+6baz38Oqrr7qssVFASksxWOn16tWLk/u1114rf/75Z2yvLbRhMBjyDyV2lsN76aWX4hF1ZHvs27dv7KhoIOWdp0xCevjhh+Pkfskll8RnoNpCGwaDIb9QYmdODLIL/5ctWzaSi22kPLEDUmo2a9ZM9tprL/cyyMs+ceJEJ9dYuKPBYMgPlNi1kCcmqotthILYAdke27RpI4ULF3YvBZ2MiBmboGQwGPIDr8VOwq+gxTaigtAQu+Lll192uR54QayeZNkdDQZDfqDEfuqppzr/XJQROmIHzEBVS12LEbvBsDNkyd/zJsqPo8bLrGUZsW17DpTYgyYoRQ0pTewbN26Ud999Vzp16iRr166Nbc1B7969t9PMjNgLHhMmTHBxwC1btox04Te+9957znkfDmyVTWvmSFrvF+X+S0+VQ4pXlfZfRleGyA3KF/4JSlFEShM73mpeBGkDHn/88R0aElkeSStgxJ4aoBPWjjbqpXz58jJ79uzYL091bJWMVVPl69ebS/UTs+//8Muk87eLY/v2HORF7CT+GjZs2A4GZFiR0sTO+qZXXnmlexmQe/369WXp0qWxvTmTmCwqJnXw4Ycfxt/VVVddJQ0aNHCzhqNSmOlcsWJF9xtZQi1062JuniXv1TpGCh16iXQeYsSugEdatGjhJisRWh2ekVjuSHmNfcGCBXLzzTe7F0LBm802wMxTi2NPHSix45xihRqkNKZnR6UQdkv+bn5jKIl9zWTpUbuoEbuH2FmpDTLfZ5993D4MSbaFHaFwni5fvny79ALVq1d35E5DM2JPHSixh0um+Gfo2bOn+42RIfatW2Wr+5slmVu2SGZmFpsiCSV2XfOUcMfatWu7bZTrr7/erbWclZXl9ocZoSB2QKZH7xqo9KyjRo1yIY98NmIveCixn3feeTJt2rTY1mhB/QihJ/Zvl+Rs+2uUdKpXVcqULiuX1rxXnunaT37dPt9eZKDEzmxT0vRiIPKZguEYpbj20BA7YGjfpEmTeOz6aaedFl8yz4i94GHEnuLYjthz2HvLslHy3A3lpPxNT8ug3+bJ0pVrJCMaebB2gBI7o3xN08tsdqSYqMW1h4rYAcOk1q1bx2egajFiL3gYsac44sR+ubz249+Suep3eevR2lKneU+ZtCr88sPOoMSuRRey3rRpU+yI6CB0xK548cUX5dBDD42/JCP2gocRe4oDYr/zv1Lo8CrS8o0P5OVGt0vd5/vLzGhE+OUJ/HGM8JUvSPfNLPaoIhTEjpW+fv362KdtIELBFtpIHRixpzgg9rtKSKGjz5DzzigpRx5yljzcd3qO8zTiQGphAWve3SGHHCIffPBBbE80kfLEvmjRIreS0mOPPbZDGFLBxbFn7RGN4Z/CiD3F4aSYE6XQMZWkfsN6Uu2EvWWvU26QbuOjnTdF4Y+K8YOFfKIQEQNSnti1IVHuvvvu7TzXEH3iwx03yPSh70v7Fk9J87ZvyJcTFmfTONgiiyd+I906tpLmnT+TBZluo8EDI/YUh2rsh1eTt4dPl0nv3iVF9yokx1RpKsP/iqjH1AMl9qCZp6SIYK2Hd955JxKae8oTOwRx6623uhdC8a6ilPAJSlnp8seAZ+XmatXluupV5IwTjpCjTr9OXhk+WzZJpqycMUQeKn+kHHXR0zIhrO0gM0M2bE7OeMOIPcWxdqq8e2c2sR9yiXQZsVZk03zpcV9ZKVyosJSt/57M2hg7LqLIjdiRdHXkz1oP/vWVw4hQaOwrVqxw07l58JSLLrpI/vjjD6e7J1KKWbtsgrz3fGv5eOQ0WfTXQpn+/ctSo2RhOe76l2SSm2W8XnrXLyulqj0rv292X0ktZK6ScX06SINaN8oNtzeQjn3HyPJYB7T5rwny8QsPyW0168u7vydnsV4j9hTH8l/lpRoHS6F9L5DWg3ImkG1e9F22sYJxdLhUf+ZzWZiK9TpB8BM7+WHatm0rBx54YHz7oEGDnKM17AgFsQOS8zRt2jQew16uXDkZPHiwW4SWz7tO7FmyYe08mTJ5WewzWCn9Hykvx5V9UIbO5/M66dPwAil1desYsWfJ5o0bsjuYDbJpSwHr7lnrZGLvJ+XG62tJg0Z1pcaFxeWgI06Tem+OkBXZN5aVsVrGv3yTFD3yHGmZtib2pcTCiD1VkSXp83+Sd5rdIxVO2F8K71dEytd8QLqnzZH0Rb9Im5qnyP77/EcKH3GqVK3fVj5Pmx77XrSgxH7uuee6JTfhEw2+wFgcPXp07MjwIzTEDtC+OnToIAcfnG11ZL+MY445JsELbfipean0friqVKn1skxyIWFr5OMGEHsbQQxaP/VTeeLGq+SmRs9J31GzJb0A5ZmM1dOkd5fXZfiMZbJuwwZZM/dbaXbF0XJQuYfkmzk5vyv9m6fkvFMqy3M/JsdZZsSeusjctFaWLV4g8+bOl/nz58m8+QtlefrG7O0bZNWyRbJg/nyZO3umzJq7WP5Oj2audiX2okWLuqRu/K/cMWXKlNhR0UCoiB1s3brVLbRRpEiR+IvRl5NoZC78QprccYe06jdJcgz01dnEXl7OqNFBZmxeKaN7tZfHmr0sw+aky4ZNmQVosW+VLZuWyYIF3oDkDBne8VopfmZt6Tslp8dZPaSZXHBqNrH/hMWeKelL58i0KdNk1uJVsmHzrkcDGLEbUhlK7N5Sq1at7I7ODccjhVAQO2Tux4ABA+Tkk0+Ov6CEE3tmuvz45lPy8HM95Q/ly8w10rfRhVKqch1p37mFNG3fT2bsGF6fIlguA5pfL5ff0kE0mm11tsUOsbcdl5k9/Jkt7zW+QipcXV/af5Qmc1fteiSAEbshVcGayaeffnqcL0gl0LBhQyEAI4pIeWInAoZpvxC5H0OGDJH9998/CcS+UZb89J481+YNGT7TQ3hZa+STRy6SosXOkrKnFJPqbYdLqkYAb5r3lTSvU1tafjJNVCGC2Muffpl0HLNMZg37QNq27iKDfk/cLzBiL3igHWtUBwbRsmXLZMmSJf8qxzjf5zf+8ssvLuHeuHHjXKx3GMFzIZ007w5nKatgkXsqqkhpYkdTZ+UkXsZJJ50kn3zySWxPDnCoJn6C0lb5e/o30rVLNxk6zRc9smW19HmwopS+6hF59bmb5JSi58qjA2ZKykW9blopP7zVTB5jROGRS9d821wqlDxHajRsLPUbtZWhixMbjG/Enhz88MMP8sorr7gIDmZM8pw7d+4s7du3l5deekn69+/vliUkouOee+5x6WdZwo+2c9NNN0mNGjWc5PDss8+6NgRJT5w4Mb5a0Jw5c+T777933//iiy/km2++cdfh+5dddplre2RRZSGKK664Qpo1ayaffvrpTlcbov1SDzQunA6GxXPIrOhdMAfwmSiVZIFza0oBLHe/CsBv4ZkOHDjQ0vYmG/Soffr0kZIlS7oXctxxx8n7778f25ucCUprZgyRztkV97XPxsmiZUtkwbzpMvqz/vLdH3/JZsmUTxtdIGfe2FlmbZwr79cvLyecdoO88+uK2LdTARtk7vBu0rrduzJu4fYVdM3QVlKx+KlyXsXTpWi5+6T/tLwb5j+FEXti8fXXX8v9998vJ5xwgrtmboUVq84555x4Cuu8ClEgSJgcz/wQViWrUqWKC0RgH9asf6F4lp9kGr53Gwm07rjjDunYsaOzfjlPvXr13AxxntGXX37ptpFF8c4773QZFKtVqyYXXHCB20Z7JVMrHQh5n/h81113OXKl00kGVGP3L2bNeg8PPvhg/LeySEzYEQqN/dtvv3Uxpjx4whtfe+01t51VbRJH7Ftl9eR+Ur9CUdn/gGPkrAvOlzJnnyNnly4lx59xq7zz80yZO3Gw1C99gOxTpJq8/cscmfljF6mwd3ZlOP1mad/nZ1m8rqCno2bJ0okDpcuL78vPC7aNI7Iyc6wTLPaKZ14mLXr2kocuPlGKX/2c/Ly94bRLMGJPDJi38cgjjzhDhmsFlcqVKztivfrqq3M9DoLGcn/99dedtc49Bx2nhQ7EH5RAYdJO37595bPPPpPnn38+8BhvYTWivO7dW5BSNbJNy9lnn+2WquM5JBL+OHbAO+Q56rV5njoBMswIBbGDn376ySXI5+ET7kgvj36YyIU2NiyZJEMG9MservaXjz/6SD5ypbf0Hfq7rNiwSdIXTZa0rwfL5198LxPmLpf0NUtk4rBB2cd/LsN/nSNrNhXkEC5L/hr/sTSt10Davv+1jJswXsaO/kH6d31Veg37Q4iDWf9NEzm3ZCVp/8t6WT7yZbmq2FFyYeOeMm1Dzhl2FUbsuw5WnkI+4RqUY4891v1FDsECLlGihPt83333uTxKpNh44YUX4sdDwlj5WOBY37NmzXLnRXrAssa6ZzvkptlRWXpy6NCh8uuvv8rTTz/tjsG5yF/2I3empaW582CJe7OqUo4//nhnrT/11FNyyy237GDxa6lUqZL06tXLdRDISGXKlAk8TgskO2nSJHfdRMBvsSMLVa1aNX49UpYsXhyNJQNDQ+zgt99+cxMJeAlUTir6QQcd5D4nSooJK9ZPHyANKp8khxx8tJQ4rZScXLKklCxxkhQ542Z5PW2hbFoxQXo9VFmOOrSIXPP8FzJtzlTpdndJ2euAE+TqB1+WYTNWxp2s/xZG7LsGtG4lGur1E088ITfeeKP7fO+99zodmEACCBdi1oynWOUcgx6Oc5PzqBGEcQKI/mCBcbbRKUD4uixczZo13TEAjZ5ttKeRI0e6c/KZ7RhSpPTgs3YgWNsQOYEM6NjIo5oCBAPsueeecx0KnyF2dHbAvTPqYDsWut4LDk5+s47EOUbXON5VKLEzQQlHML+Bz5SoLbYRKmIHEIZ3SKlWxZ5O7FkbVsic6VOzn89UmfzHJGfpTJr0h0yetUTW4ZPavE6WLZgjM2fOlLmLV8r6jZtl/cqFMiP7O3/OWih/b9iyy3H4Ruz/HoTjQZScG0LEsp0xY4YjPeq45g7v3bu3M2rQvJEqmP6u5I++rcDaZRtWNMBhiX5Mh/Hxxx+7bfirOBe5ySFttGYkFFYl69q1qzuGCYGcB22c94v1znlwnjK60JWIcO4qmBWu97NlyxbXQUD+aPjIqqBfv37x+8GCJ/IG5yzSDLIPjlydVY5Gn4gIFpVzuWapUqXc//xWOssopBHwInTEDmhQajlo2dOJPRVgxP7vwSLZOqNaSRyrks/MlPzuu+/ctq+++srJLJA918fPhFOS44hWUSjZY4kiw5B+g8/IECpvIK+oYxZZgin1/H/iiSc66ROQIAurHEtaLXQsXzoi8Oijj7rvYNFjhWM4QOI4V3H+AjooHWlTR4Dmfrr44ovdZ8IxieZhW7t27dy2Ll26uM90CJ9//rnb5ge/jXuhAyH6BoImnTfPhRBH1krm3BQlcy1ISlwjKql6vQglsQOGdOh5iU0pYNgV7EnEDpn+m9jwICCTqCwBuakkQGiiPk8IExDWqFFi6OJowujuOCw1qADccMMN7hg6IM6HA1U/Q3oA8obE2Y6+Tupa/i9fvnx87QPCH3GWorkjE2HhEvKo6NSpk/sOHQnPg3vC6sayh9AB9w6BcxwjEYCuz2ckJgUWPtsaNWoUJ1vttDieeyG0k3P06NHDdToQM3o9zmE6NkI0iXBB8yfKhu8R7okhqB0nBb+AN8Iuakh5YqcH5mVicfiB40gdOfkldqwcNEgaBN/Xxok2SUUnaySdBvu0gWGJaCyuF9wbVhVDTQphWlFI+flvsScRO+GCkAuW5K4WNGgkEQjxxx9/jF0pR67gWhCtjg6GDx/uLHi20yaoy3wPTdo7z0MtdnRtOg4ldshZZQ0IUsmOevz222+7/yFhre9MDFQHLhYvxE4UiYLwRPah31P3iYHHWkdGwbkLGBlolIxa7OQ+5zPRPwokEbYhI+l38asRjsn2RBVGFEg9fkSp7aY8sVOxeBlUDNUGFZByfsMdJ0+e7GJuL730UldR6MWpQOh3DP2wZOgk0OGuu+46ZyngJUdjbNCggTz00EMu5pYJIVRAKjSWBY0MjZIGgiWFxcVEkl3JFMewmAaP1khjpdPIS2NkuMnvYzj6T8F5E6Uvei3MqBN7MgrWq9eBh0HDdogdAge0B+o8I1UsYowQ2gZk2r17d3cM9UA1dhyrODWx5vkMsev7xrGq8eljx46NEzujB61vtA22URgVIIvgEFWoxc6ycxhFTHRSYtdFcTCY0M85Ti12TQfCNRXo+notrHMF9Yo6hbOzYsWKrv3SXulMqlev7py/WOZ0OFjpPEcCKx544AHXcZDFkTBN7aCCVlDiGrRdOp4gIy5sSHlixwLQoScvhhVOdNZYfiYoobdRqb0e8GQU/wQOKi7DQxxS+cWwYcNcJASOKiwjPReNgrwWNAI6JxxV6LBvvvmmI3+iHPh9HEPjIBSURk5lffXVV902OhqsMobKaJ9YcMQK0wjq1KnjnGwkV6OB4xSDNCBnOozff//dSQDjx493+5nAQWPFEsOC5JwUzsH9GrH/u+I3XHi3bPcSO8dQ13AAYokzmtRoF2QIwPvTUEKOp71QX/h85ZVXxomLeqAZUsluSNviGK/Frho7xzEyoF4ifSj4n+9UqFDBjSqIjilcuLALQ9bQQeoCv4Hj6KyAtmm14AF1m22MDKhrCgw4jB3qJAm76MxwHGPQ0L5V798Z8C9wfm8cO8+GkZfG5uMMxpEcdoRCY8cKoJflwWMdYy2DnU1Q4uVjVbM/qNDTo9NjhfBZz8VQDZLU4aK3MC0ZEtVoHAr6IxYCs2RbtWrlwql0H5YTBIqVg/VEnPCTTz7pCBzHE84t7hGCVauGwuiB30wDUD9CUKGBB22neDsbGgtTqdEW85pgwu/D0UVDhKAJm0N24F5wmnEeOi10XWKrcb5hMVJozDyXPYHYeQ68UzrNRBW/MxaHKdfCcTlmzBi3jQ4ZcmWkiNEAMWm4IzoyJAe5akSJOkoxAviMNa7Exf1TdyE8ZAiMAY7h/eN4BGqRUx+o9xyPVQwYGWARs5/2wmfaqkba6LXp9KmnjGoxBIC2ESxpQFvGwGAbFnYyHJr+CUqMXNq0aRNvJ+xH/qKzCDtC4zzFSoQweAFUHOQO8kvoBCWGZH6ox16/A+mgPWrMLJUKqQViL168uHMeUXEhU6xdnZHGkA4Jh/91Gjb/c5zqf16tECsX6zUvQs6tYGlhkWMNM7TFqcW5GAZ7j4NYNHzLW7CWaIRqEflL0D3RgWHJ+bfnt/BceYY0kD2F2JHqlPySBaxU6jfvFAkDElenKCSu10cvZhudMo5K5BPeA50x1i2AUDmGDoGRIbjtttvcNiJUIGUImM902oQfAuo+22gLSrzIIAAdXOPu6QB0G4YD9QzpEminQr1UJ7C2QW23hGOq8aaSUqKhxE44JqN9nKw6MkbiQfKMCkJD7ADnJM4gXgQ6HhVNLVa/xY5+yNCRfYRp0UAgdxqmWiGQsVYwZvthMUGgVEzyWCCJsE8lEv6nImKxUnFpQG+99ZbbTkNXhw9A18SSoqNgGMtQlSEujYJj0fi5puqC3D8VTSMJvMCq5xpYVnyP//H6I/Xwm4gHphPjL5Ya8gsjAI7DMtLZikQzNG7c2FVmiFg7Re6DoS/6LUNjYpQhC2YZYv1hhY0YMcI59pBgSCJFR4u8gzSDxMP3VY9NNLFv3ZolWf8kyN4dv6tR+cFQYuc9+i3sRAOLHH8P18PPg1Si9d2brIoOgOgW6hqdvb5XRocqqTDJR6UIrFTen0bEYNDoMYzAaAMYF0gp2gb4rLILRgWgLlDnaGfe4Aad1EQdZCKUtlnqt/qBkHjYxigVSYU2xnUZ+ZH0LBlQYkfS1Q6SQpuKmiESKmIH5HFQK8FbqPBYLFoYCrIduUV1P2QIhrc6dIVUdUiIQxRrAsLjWCo0x3MerBe1buhQ+AuBUrE1EoRKgxbtBUM9Gh2aJ5YTIwyGwUhEWAxopDjLqNhs05mEXjAsREKh0TLC0OgGJCQ6GxoVv4HrMGTnMx2ZOs+wlugAsOBoYDiZ+J+/aOocw/A8EYsNJMd5ulZmjx0krzR/RBo+8Ji8PWKJ5DZI//Prt+TpxvXk0TZvyxdj5yZl4ZPdSewQtxoO1Es1VNDUvb4bHJ2M6niv7Kdg/XrrI4SqUSdIIhgb/A/Za24U6pqORglXZBTL/0iUtAFGj7QzjBqMIzoI9iPdaOoCgFTKvXDP2nlgLOEvU2DAIDdiZKD/U0c5DuPDayAlEkEjXIzDKEayhY7YASSkluvOCkSOPsn/yBNYllgyfCYiRi0SLF4qGwRKZYQ4VZZhu+bvYBt/IX1CsRhu8hkZQvNpJBJUcjoXGgCWNLP09HoMbdnHMJJ71MkmdDpYVdwrE1qI16WzokGii2MZIWXhPON4KjyNdlehnVxiiT1LNm1Il5861JTj9s22tq7uJFODJNB1I6XVNcXlP4WOlbq9JsjKDclJAbs7iR3gOMQHg0EBWdKJq97uBUYCZIusRgQXHb0fbFPrmUJ9IfpKgxEAozENp6RwXYiaTobjGCnqPm0LkLL3HBAlI1uVD/nLvXmdnJxPpU72q0RI5+M9V6JA54fBxzX0mjibNaY/aggFsWNJ+J0pWL+33357vEJAdDj5/IXhJR0BLxKnKJEGSuxMiNAZcFgdanFiIeNYwsrA6mDYqtO91aGKRYM8ocROg8EyTzQYRWDZ0MCQRzS2GasLAsWa4t7RKFXbpGNiSMswmWEt8ovXmsPq4jzaSSDnBM0T+KdIDrHnYGm/h+XyMkVkr4PLydPf+lNSZsrsT9rKzWUOlr32O1dajkheIqfdTeyAHDFEt0CwO3uuSIB5ESP33Lp1axdJRf0NAiNc5BTSGTCq8xIy1j2GBHUIowJJw2utK+iQkASRHZHogsJ1GVGork4dxH+l0T+JBoaLdliMfIjASUYHkipIaWLnwRNmh8ZMg/K/CDRAlVyCnKcKZA50ZsiRWFWIkO+gG9JgqFQQuG5nqMt1NUSS62v+C2a3YcVDoJA9pMl2tHpvDHKigDwDUXN/DMu5Z66HtcMoA0seIkX/pgOA0LGO2MZxWOxYJvyveT6o2L169XLWGtuxAv0y0r9BMol9Sb8npXHrp6TyiSdI6du6yjxvP79ujLzSqo08eUslKXXSefL09wuyqT45KAhiLwigzecWHUK7o91gaNC2dgVEziAxUneSaT0jgWpAAfJT1JHSxI71oZEtODywHryWu3eCUl7EjtatBKwFpw2yDPuQNLz7cOxwHRxW3u2MCpBf1IkEQTIKgFy9Ey0SCe4PWYVr0MGoUxSyRj9XhylSEZ0NVhbOTk3OREw7E634n2G0dl48Dw1vI+InERn0kknsi/o0kSbdf5APHr5Uihx/kbQZuW2RkIWfvSDNOveUnh0aSLnjS8uTRuyGAKjGzl8/GMHSwURlYeuUJnYcPliVuggtJM7MT/WsYy0ose9s5imErA4jCg5UPQ8RBmjtECjkiV4JsIK931GvPuGMel0KMkyiUov6wfU0DSpSC1Y3HQokiv7udwgRI08YHDIVnxmp8BsgfaJeNO6Y7dqhQfJ0oruKZBL7gt6PSdMPfpOZ43pKtf8eLuXqfSQuGjtjqnRr3UJeHTxDJvdpJGcedaYRuyEQ/jh2BfUWnxn7SO+gkURhRig0djRuzS+NbIKzBt0PmSK/xM7x6Mjo6jgO/UQMEeGUwkJWYLUz3IQMsdJ1nUbIltlqWLqEo2nmvWSB82uYG4XRiU4yQXZR/Zwhpur8+BI0godCtAGEz7P0bmcklKjwsuQS+yPy+PsTZfna1TKwwZlycLGr5Z05WyR9VA9p0aqTjFgpsrhnXSm1hxO7P7IKOWVXiAqnvOZCYnZqmOEndjgBaZNRN9sx7Gg3Ruy7EWjeOG14AejN9KxofWpR74zYkwEcPbsjVIoKCKEgrzDBimehIFSSGa84qiBtBdtxEqPPE0OvETucC2cWEQ2aokFHLruKZBP7o91/lVVbRVaP7iDnHXmcXPfE69LjzfbSrtsoQQ2e2e0eOS1ixE5nTA52jAtmNeOsx8hAoiTcldWTeO+QLjM28Q8RmkskDZFdTDzDgck2jmXuAUaNjkoB74rcLIzoyJeOIYHxQ13Dz8S8DfRpghHwUSFZ6KzSvIDD1Ds9H+mU/DFBurw/OCIZ8E5QosPTeSBaZ3NzJocRoSF2gEcey5MXQZQLFVdfTEEQ++4G0QfehrIz4IxiUpc/WgfZBZkJHwNEnygkk9gX9XlUHu8xXpbjP984Q96pfZrsd+ARcnHDDjJkQU7H9GePe+X0o86Sp4YvSkoMO9hdxM67w2/DpDav7IeDXMMMtRQrViwuV+ZVkOOQ4CBo3hESHx0Blip+Gq6DJEEYpDeKyl/Yx3eQJuksyMXOfBHOR3ABYbRMbqOtQqJ0BsyU5X9Glcyu5roYGHRahC4jZ3IepFfvqDmRUGLHv8aMWvxWfGbknawFtAsKoSJ2kFsM+55A7KmO5BD7Vsncskkmv3mP3NPxO5mzhpWesmT2l0/IKQccIzd0TBMC6Thm7Is3ygn7FZX7B/wpG7YkxwLcHcSOBU6qCp2Q5C+EumKFQ5Y6qziocA6kOhzoTMTRGalBBbJWovMXLFuID4ve3/aYH+H9TOE6eV1LCyNvjWrTgixCZ5YMeVOJ3dtp0ekEhWuGHaEjdqAx7PpyKEbsBY/kEHu6zBr3hbS+raKcf00jeWvQr8ISEJnzx8j77Z6RD6cw4lgjM8cOlOY3V5Bixx0vlf/3kgwalxxndrKJHWuXFLhcA6KlQESEpGIB4x8hEoqJRMhtjMaYRczxdARYvZAv32HOBW2FMGG0cg3ZZZKadgg44xkZIM+QHkJnm+LT0dEw50dWQS7RmdreQuABk6KI2NK4dC2MrLkX7hnpCD8Xz7Bu3bqO2L3HegujBxbRSNSIkmegxK6FaDH1m0UNoSR2QMw4kyx0WGrEXvBImsW+OUPWpafLmvS1siFj87aUAlmZsf+3yhaOWZsu6a6sl4zNyVHZk0nskJ5m+EQugQghRaQW0lfg02E7+3HyAdoBhM1xpBXAX8JMYw2D1SABZi1DskRWkRCMkFjmPNB+ODdgG3INZE/QAOHFkC+dCaSPA5XOg3MTOquztnHmE6HFtTmG63I/7McXhvzCZ29IMJPjGHlwT/jONLqLuqOzY9mXqIRgdIA8R87Lb6IjYuJXVBEKYg+atQbQm3UoZ8Re8Eimxp4qSBaxcy61hnFYYpFrfnIsdY120ZWRCMvFCiVNAKQJmaJPA0gTSQOZBEud4zRLI1o4oEPQjKWkDMYyxjnLZ7YDHJ0674Hfrfu5T8D1uAYdgUZW4cSns6CDIrAAvRzrnO9BpmqB6zZInPZNZ8RnOilGLaT74DOSjndVqX8L/HOatwZ/RFB4LzJvVFIMpDSx06NS6XDI6BqMXngnKBmxFzyM2P8diNDQdQMgMqKeCLnDEmYb+xQ60Y6ZxxzD9H8+Y/FqOCJzNjQvCpEeECeEzmfIVaGJ7bD0iVTRrIx6DNJLrVq14scQXcX/TNwD6O4q3dCZAJ3ljAykwMHKNvRsrgOpYsWzjQgbwJR/orSwppktTaeiIc5IS/8EjBzoUCBpOjC4Ay5RB3PQzFP8Gmj7POsoJAVLaWInlScvgoJ+p7mlFflZQcmw+7AnETvZMxM1lEcaUWcjjk4AARJ5gvWL1qzQ3EbIF2jsSCZ8hgR11iTRTnyX7axYBLlpqDAT8xRqFTO3AymFe8ACZ46GArmTY7CwIV7kEd4zwArWSW48F6Cztb0zwTHO2EYbZZRN3UAS4lyEVwLWHNY8TCrZIDeh86P3sw1Zhr/MuOacZIWk0yDLJKMJOh86Fjo/nhP3zG+kAyP0F+mJ8/snKDFC0OeF7OQNGw4rUprYSWylWRUpOHGoAAoj9tTCnkTs6LWE+6Eh72rRxVogSTVesBpJw4zzlKUdFd6MiJAx8ed8hti1bTDRTtcUJbEd54K00caVgIGuRQCh0taQNdG9dfk6oCksuDe+z34lPlIHa/tkwRqg4chIRQpdlg+LmVnbjCJIzwFpaygu/gPSR3McOZEAlj06f27ROv+k4Czm3vkfJ6oCCUizotJx0WFEQXtPeY2dHl5XbqFgKekqLEyyMGJPHexJxJ6MgiWqE3UgeIgdi53ZkQolWp4xUqRa7GjTarEjkagUgz5NG8LpCXF7SVstdmZPo8UrsZM3RaFaOOfnHOzXtMFMUlJnLlYz0Mgb/AMKTVzHPtouTlqIHUtcF7yG7JV4ldgVaP7IJ/ge0OSRjAj35L4x9pCLiAYiKR+dLSMQrHeseO6LkE9CNdV5SieIjMU9Kn/wrL3PJuwIhfMUAlcNkoIWxjAQbRLHDduM2AseRuz/vkBK3tmgjEbVuUl0jELDfNURqRY7xKc52CFmtYiJ0Wb2KqSGla+dBMSmqzOhc0P+TE5C41ZixemqVj1OW9oa51RiJ1KG0En2swAMUDmFmbEKXZADSQQnMH4BJXaN2sGvALlyHB2cgo6OuoSlT5vn9zAyoUOg82PUwHNDluLc+BPQ2Ll3PzTyBscuIwqNwafDiNKsUxAKYgc4QpgyTTgWLwNvPT0/0QB8NmIveOxJxA4ZYC1Dbokoun6pApLS1YyItwbEXKtWjmQAsJqx6omCUQemrpREuCFWPY5ElUjQoAHr6ZIJlG18D9lDSZllIQEEqqGHpDRAssBqJ2IHMDLQVLiEUwIIk89INwBZg6Uf2UYHBeEyK9VvsXM/6twkpXQyoHHsPC/+UpgBy7WjhtAQO6A3xrqgcvFSCF9S/c2IveCxJxE7hBCU8yRRwKJGTuBa6ojE0sXpyDaNPYfsVf5o0aKF09OVoCFjrFfOpbILUiZAmlDHpL4rDaXEYQq4Hto8xhSOTCxeQiuVeHH6cg60d3WCkrcIqx/rnueD9KKWOHmJAKGRSCBYzLpoNpEwXIcRQDIWlcby96ddwF+BnyKKSHli9w5PAS8IZ4xqY1qM2AseexKxJzOlgALyhUhJdAcpqr7OaJUEeABjh3BgtnOcausQLsSrgJghTo7BQmf0y3HIPbp+Kh0D29DNkT1YRYzPvE+cm7qOroZf6loJhEFyPOA4roHswyLZGp9OWKQ6XSF8nV2rET+6EDrtOBmzQZG21KGMMcjIwj9KihJSltixNJiJRwwrnn0/CMnS8CWtEIaChRF7YgHB+RduR0bo2bNn7IgcoE97p/IzoiUM0DsJB0te49BVf+d/sjjS1gCx5JrCFitb9XNCBTmXEj2WL9EqSDPcD9dSILXg2OQ4jC8dYeDg9E7+0REE/jLaucasqwyUaHBfKsVw//5UBUy2YoEdf0h1WJGyxI4zhJSjvAicOgz/NGJAQWY4dYAYsRc8jNgTD2QLtTSp68RnB0lA6MQ4YJGI0N+RX7yAyDCGvEm3aF+EOSqw/nV2qRZkHn2XTHzSyBItzEL1ngPg0NWJSxQ6iaFDh8b25gCdHQnHey7Orfp9MqDE7p2gBOGzzJ+mcmDk4Cf9MCJliR0SpyfXIRtOUnpU70PHArComNSBEXtygIRBhkXq/67o+pA9BhIaOgQWZJ0i8SCxINEQSqh5/BUYU1i8TOSBKD/66KPYnu2BPk+oJI7ToIVcGCXgQ9ARAqMD9PlkQoldJyjxPFiRTe+BjoV8996RTliR8ho7FUtjZclFzbBPrRGboJRaMGKPDpBucrNcieKBAMeNGxfb8u8AuRNNQzgkEW7JhhI7oxD0dfwMOHrZRo743XEPuwspT+wAz7lOXUY/ZLYevSohYUbsqQMjdkMqQ4kdaVcXeKcQRWQLbRQQ0BB16jUedxZtZnIDHni2GbEXPIzYDakMJXZvIcTTv8JYFBAaYge8AA25whtPtjh1wBixFzyM2A2pDC+xE0aKI1pDPaOGlCZ2ImP8Dx7njk6k8BYj9oKHEXtYsUUy1q2RNenrZOOWbVPxt2Zuloz1a2Tx/MWyal1G0taR3R0gCkdnqSPnEv3jT/YVBaepImWJnZwQxN1iofsbEJEBeNuRZIzYUwdG7CHE5pXy80dtpM41FeXcMufLVfe2ka+mpbuVqdYv+kW6P1ZNTi19q3QbmzMhKqwg9p/4fd4dk7g0dl/BTF5i7Ynm8e8LI1KW2JmxxnCJF0GeC3+sLL0tOTGU3I3YCx5G7GHDFpn7RXupe3c9ad6hi3RoeoeUPvIgKV6tmfywIku2Zm2R8a/fJSWKVpAXfszJ6RJWQNaaUsC/0AaTkzTtAcZkFCYppSyxE46Eg5R4WR44yY+YGecFUo1FxaQOjNjDhqXybZ9PZMwMlTvXyeiuNeWYo8tK08+WuS3z+z0mZU+/VDqHnNiBZnfUOHbCOUnVoDPYCcQgS6XFsScZxKsT46p5mok/1eRHwPKxpxaM2MOGdbI6fSNrgsexeHR7ubTkedJkcM5SlPP6QuyXyYs/kSwrQ+aO/Vb69/tUhv82Q1ZscIeEBuo8JX0BKb9Jp6DBF2Sp1OyYUUBKEzugVyWHtM4wpbfVTHJmsacWjNjDjgz545MmckWVhvJlTI2YGyP2LmNIxjddOt1SWa64p4V8+O0vsjBkCw0psTPDlHz2/E8hLUIiFsxOJaQ8sStoUJpQiGWu8HIzQcnysacOjNjDjcw1U6THQ7dKw24Tsyk+BxB7uTOvktd+GCfD+78und4YKFNWx3aGDEFx7LxHctpHDaEhdsDKMORg54XQ67LwL2kG+GzEXvAwYg8zMmTyp+3k4Va95M/1sU3ZmNeviZQ/40y55IpL5LpGr8vvIc50qxq7FvKxR2/UlYNQETsgIb9muyMeVVdDMWIveBixhxfLf/1Y2rXtLj8v2T4/zNx+T0j50uXl6ovPlFPL1ZBOP4QzYoS8NqQS4N0RkMFKUruSUC3VETpiB2SLo/cljamGRBqxFzyM2MOJtdO+kJc7viPDZ8VE86yNsn5tupBqb9GAJ+S80ldLl6++kM61y0jx8vXk01kbc44LEVgpCQmXd0doIzJulBFKYgck7fnmm2/ikw6M2AseSuystcmSaFGErggUCWLPypCl43rK/2pcI3c/3k669ewlH7z3jrzw1JPyTJd+MiszQ6a+fY+UKFJGnv1utqye009uKXGoFLuqhXw/Z5Ws2xSuvOWqsfvj2KOI0BI7INZdo2JYcd1QsGApN94Fq/moQ4q8+oksREkx2SRoX34KCytQbxiGcx4+Bx3DohOkruWvHgO6d+/ufiMLWpA2OtyYJb2evl0uKl9BKlW5WC6uXFkqV64kF116p3QeMk8kc7b0aXWXXFrhCnnk9W9l0Zbl8uUzt0r5C6rI3c/2lglLwjVDU1eZ0jj2KCPUxE7j5EVRmDFGyNKeUpj6/N133wXu08Kak5SgfckoLVu2dO+CHNes5sPatB07dkxoIfaYJdqC9uWndO7c2a0dygivdevW0qlTpx2OYRuT41i2jb98h+2sz6l5iphT0b9//8DnEJ6S5mTNEWlpMiL7L/+7MiJNfvzpJ/kpe/8I/s/+mzZihKT9mL0te/uPaSPi3ws+b2oWXf3JiD3FQVqBOnXqxFdpt2LFipWdlaJFi8YYJLoINbErhg0bFvgCrVixYsVfyDEVdUSC2NFzNZ6ddRoZSjNkJx0BhTUcNXqGSU5PPPGEtG/fPn4MKTxZcV1ffOXKld2wW/ezSrsu6MF5yDhJDL33GpSGDRu6UCqO42+DBg3cEN57nPdanIsFhf3neuCBB+JpFPbdd18na/jPc+2117r9FO6tRYsWbh1L3U9BGtFIAAqrx7O0oPc8rKVJtjs95rbbbtvh+bHCjO5nGIuE4b0Wv1OXGON3s6am935ZgkwnklFq1qzp5A3vNZ5//nnndNVjcHBxDe8xlMaNG7tnwjFMB7/nnnt2+E262haFNXPZ5382N9xwQ/wYcoXwrPzHMDtR3yfXQsLxX4vCdzXfCEnpOLf/GXqfMXmPWGvTu5+1BbSOYlHyjL37ve8bSQFy8t8vz1WP4RysNOY/hrqlAQfca1DdQoLS8xDhxDbvft6n/l7umTrjfy68z0qVKsXPc9FFFwU+O+/x+v4PPPBAefDBB6VDhw7bHU+90mdEfXrssce2a8cU7k0Xpqbcfvvt2133tddei7FGtBEJYidChorMi6QBLF26NLZnG0iqz35SE9Do/CBPhM5sZf1DluPzgogPddRCdP5skwqIcu+993bkA9HjePMDJ+PRRx/tzkUjCjoXDVLj9CEUfypRMtBpQ+a+6IiCwIrxWtHxQ/z555+xPdvAAsWasoFOy58Df8aMGfFGyurzrFPpBb+RjoVGx++mAfpXySdMUAkFx+PChTsmlRo/frycf/757hg6pNxyd6CxcwydCdfyh679/vvvcXLntwctuIx/5tZbb3XHsJgxaSuCACHwPvVaOFODwDOkM+YZ8F78jlWvxkuukgkTJsT25GDZsmWO3NlP3WBNUS9w5NIxsx9SZWV9P3hv3nO89dZbsT3bg+fH76Fu8ZvUMayYPHly/PmRERE93Y8ePXrEjQ+MKdJ7+PHbb7+5tsQxOC75nBe4X46lM8X48oPkXBgRHKPrHweByCU19PCJ+POu7wmIBLFTqbAA1WpkhXU/WdJwtCET/05P76/QOMPUMmfVdMIpvfCSExV21KhRsT3bQHKhJk2auGMo5I0nWZkfrBZ/2GGHuWNo6P6FgTMyMpzFpWmJb7nllh0IF3LE4mc/vx2LJAh9+/aVY4891h139tlnOwL1AhJm1KANFRL3hyvSeapFBRFy/15AroyM2E9hVp+/g8VaUguYa0AgfowePTruM6HDopH6wf02a9Ysfi3yaLMAixfeTuKggw5yWfv8YGlFtYR5fnSmfnAtOi29Fr8Lkg3Cq6++6joBjsNq93dedLL6Hhj5jBw5MrYnB4RPYs2zn3eB49sLXaOA/Twb3qsfXJNQTI7B8g363evWrdvuXbH+p5+Yf/755/gi8kzswVnvBdFJWNQ6QRByD0p3S14njAGOYfTE+80NtNG6devG7+vhhx/eodOmLXEt9lOXqLd+o4fP3Ju2nZtuuinXdxZVRILYAS8TYtOhGpbnnDlzYntzQMOpXr2628/Q+o033ojt2YY+ffo44uIYMr4ReeJFt27d4hnhGKaOGTMmtmcbaDgMJTmGUr9+/UCrgdA5rHKOoUFjaXpBpaZy63noJPwNkN+oQ2c6LDq4IGAB6ogDwmMNWS8gMIbDSrxVq1bdgZj4rdrYeUaMPLzgdzNKYT+FRurt1CADhv1KBlwjKN4dwtPr0Pn5OxFAx+clp3r16u3Q8dFZ0mmyH3InRasfjEaUCPntSCR+0Fkz7Ndr+X+Xgt+HRKLHIXP43xeWthoHkLR/EWXqAKTPfkahgwcPju3JAamrdT+SXtDK+hg1F154oTuG5xc0YiHk00uiSFw8Uy8gc53lzWjD3xExesGy1nM0atQo0HJnNKNtiro3ZcqU2J4d8ffffzsjhmMhZkbAfuKmQ1aDhg7QP7oB3Ju3fmDp85v3FESG2AFDNaxcfZn01Mw48wI9Hg2d/Vg0SCx+sE2tabRR71AU2eHNN990JMp+tEO/BQxo+H7rwz+M51xYeaoZQzD+kQaVkY5Bz4Ok5G+AzPJUaxrSoMOCZLzgWmxXq5zOgO95pSLuD0tYrU6G9VhRXvAsVC/GkiMZmxd0YEhH7KdAGN5OjUaKfqwdMNegofqBQxydnWP4TZCDf4TFeXVozvnQxCEGLyAjRikcw/tm1OU/D3WC98gxdLRdu3bd4V2tXr3a6dMcQ+G6bPODe9JOnQ6M5+knprfffjv+zpFXsMS9GDFiRDwnUpBljxXs3e9fpwAgJWrcNtINi0n4wbNS65d7xf/jfzYYOqqn8z78cgqdudZz6g1E7M9nzjmp59pmCDUNeucKRl+6tjGdbZBMRgfoHUEyGvIDw4ioOY7h99HpRmn5u7wQKWIHvDivdXXnnXfukBMCCUUrPU6YQYMGxfZsA1aAkiAVmvhdJUH+emUFSNJveQGGfwzd9V5w7PgtGrViVUaiQvvJHb22du3a8fPQSfgJDM1WfxOWOQTmB/eNRa8NDKvHb3ny/LwpTXGM+Qns66+/jg+vkawgYS84p/93e89BQ3/qqafiljsOLr+UAoYOHRrvRCB3pDI/8fBucaDqtbAgsbC9gCh11iGjLUYA/o4P6570rRyDPh0kGdDJ6bU4Bt08CN73zm/kmXs7UH4D+rCSOxaq//dTJ3WERadD/fL+dix/735/5wDohNV5zuiT9+TvZJBPVLvn2TBi8csfjHRUs0Z79/tpkNzUGY2Vjfzjvw6/H9LXd4784x9hecFITv06XDtIkuP3lSpVyh3DiCJIGuWdMXmRY2hjdNp7AiJH7ICGrZYcBSvXPwyDnNTqgZwY0vpJA3JUcqcCeeULiIFGoLIM5O6XUsC8efPiFYuClOMHFri30gc5QmmA3kgOLCA/kEq0IaPzBwHixjLDWQpJBQ1PadjeUQLH+RshDlRNqoR26newcb/qyKPgbPOC3+yVmbDy/SMRQNI3dToSnRR0DKTI6IxjGI0Fab28X8iNY0gAFSSNpaWluc6RIXxuw3beJ/cKQfg7By/onC+77DJ3PeoI79177/zv9aHwvLF+veCZISGxHz8CcpkXTADTThpyDnJII+VoUACylH8EBiBR1fYhUX9HDbw+LH6XX+aE7KtUqeL2c09+CQlw/16Jkjqa1zNkdKCSHL6JgQMHxvZsA3Kg+i04NigFL/emnQQjcTpFf8cTNUSS2AEWoteSQxLwO1A++eSTuPYHOc2fPz+2ZxsYNkPukItfT6dyYHlphfc7uxTouKzbSsXzrgDlBUTDSAPLHPIIAttpwEgLw4cPj23dHgzTGf77LXoveDY0iCCdWMH3IRN9fljdXqsToN2qXkxn4QfPEx0dX0SQBcxvRhvn+UHMud0znQjD6LwsPCxWZokGyRIKyIbQxCDiVyDd5SdVgP9ZBAE5RImJkYc/twy/F6MDiQj/if++6ITxe5C0KsgfBCnS2XF+6rHf2a+gg0ArZzTjl5gUjPgICMCZ7Cdthc4sRq4Kilhi1MPIiFEwo6Ig8JsJYuAcaPK53Y+C9kLb5Lr4HIJ+IyMKNbDodII6f+5NpT3uL2iEHSVEltgBRKCOGF782LFjY3u2gUrPsBqrOjfdD8s9t2G3Nj7INEhzVWA1TJ06NfYpGFRy/zDYD8hhd1VKOsIaNWo4iwhiCSIzOj6kG7/cpYAk6NhyA++I0YffFxIVICchlRBiG0Q4vE+IKcioAHS+/tBbL5D2MAiCRnAK6ihRLjuzUvG55PUeqJ9Y2UiHuXVsGBZY1nl1fNQJQhuDHK1BwI8DGWMABI14ASGcdG4YcLmNthilE/6KLysoIitKiDSxAxoMlieV0T+UVaBn7ko4VH6st7CCRuh3kPqR13DakEPeQaRuyD/wr6Dd59bWeL50KDszjIhymz17duxTdBF5Ygd5WdIGg8EQNewRxG4wGAx7EozYDQaDIWIwYjcYDIaIwYjdYDAYIgYjdoPBYIgYjNgNBoMhYjBiNxgMhojBiN1gMBgiBiN2g8FgiBiM2A0GgyFiMGI3GAyGiMGI3WAwGCIGI3aDwWCIGIzYDQaDIWIwYjcYDIaIwYjdYDAYIgYjdoPBYIgYjNgNBoMhYjBiNxgMhojBiN1gMBgiBiN2g8FgiBiM2A0GgyFSEPl/ZJ07zKJoWgYAAAAASUVORK5CYII=)
physics-General
physics-
A pendulum of mass m and length L is connected to a spring as shown in figure. If the bob is displaced slightly from its mean position and released, it performs simple harmonic motion .The angular frequency of the oscillation of bob is
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)
A pendulum of mass m and length L is connected to a spring as shown in figure. If the bob is displaced slightly from its mean position and released, it performs simple harmonic motion .The angular frequency of the oscillation of bob is
![](data:image/png;base64,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)
physics-General