Physics-
General
Easy
Question
Coefficient of cubical expansion of a solid is
If the temperature is measured on Fahreheit scale, numerical value of coefficient of linear expansion of solid is
The correct answer is: ![0.000005 divided by to the power of ring operator F](data:image/png;base64,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)
Related Questions to study
physics-
A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is 4/3 and the fish is 12 cm below the surface of water, the radius of the circle in centimeter is
![](data:image/png;base64,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)
A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is 4/3 and the fish is 12 cm below the surface of water, the radius of the circle in centimeter is
![](data:image/png;base64,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)
physics-General
physics-
Consider the situation shown in figure. Water (mw = 4/3) is filled in a beaker upto a height of 10 cm. A plane mirror is fixed at a height of 5 cm from the surface of water. Distance of image from the mirror after reflection from the mirror after reflection from it of an object O at the bottom of the beaker is
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DxT39lhzaBMcWXsKRqB1vWLeft8ZOZO38qK1esY8ficeRnF+EdXYI/0jjpzYAQEpqwk5Y9ldmL3PSueo4+LUoiDgN129gTS6No7tXcfEkOnfkTaImAovVgq65i7K58Ji5ewpyOEE37WkgfP5cFlc3se2kdfeEwISBNg1Fd++jilILqAgFCQVXAnunELkkIVSHW38eAKx2R4cYhecifNo+chALxw3g7ZvL2AQeVxXa0DgdOdzFlOek4aUSTZWRJGl0y/w5w6g7bSCehzhp2pI2nQHJgt9lwBHKwtTbSdqCWbkVAoo26phYON4WtDjRNgBAIvZrQMsjEqG32d4ETz2iaCqF29mxayZsbDjGoSriyi5h4ySLKPU5sNjtZFQu4bFIHr299lgcOv0tmQsM9fgaVeYMMrFxBU6SEvXtn4129ioNtadg7ZlPWf5i24HbqV++lamwBgQIPoyr30cXJ+dEMt4WqKmiOAL7iGZw3PguXQ0Yg4/KXMeuyiwl5N7GpthscBYwtKqQ0r5+mcfO4cmIRYxwe7CXVLBiTQU5hgDTGccHNCVoCGbg0zXJ3jOKjiZOogtJATRCNhAmGY6jChs3pxut2YJPMyhgFTYkTCUcIRRMIYcfhcmKXFZRIhLiwI9vsiHgIRdiQHG7sWpRYLE5CduFxuXDYJMvvpt9T48Xnn+PZZ57i17/9Pf7MzL/hMIzib42T86NJdpweO85jpOwLISNsMh6fC49v2EVXWtIv7qSfHbhOtwTA5N0wSFxOhWxkFP936OR8KnDsarDhOKtTuUeCyZDY3tFBb28PY8YUkpGRMapsZxDRaIS21lYUVSU3Nw+v13tCRTsnn05PTw+//PnPuOnaq3l7xZt6vedoGd4ZQ0N9A9/+5te547ZPs3HDelRVPeFnzmpFE5KENEJluqrpfKsD/f1DfK0fkox/j1CUBMFQhHA4QiKJ4/d4OCsVTVVVerq7aWttob9/gN27d1OzaycRg8gXMBxwAqGNblXPNDQNxAmYwofjrFW0Awf2s2bVKg4drOWnP7mP+370I7q6OvUGo9PXOYezUtEkSaKoqJgZM2cRTyTYsX0bh2prSSSUD1u0UZwmzspdpyRJjCksZEJVFXa7A5/Xx49+8hOys7P1wHuyv43kwxU0fbYbtt1OPjCCYYQl1nX9hxO3Sepj5Oup9xFJ90tdYsyfj5ZV07SUz43Uz8m0OVVZj9mGIW/66TIFnJWKZmL+/AV87d5/5Q8P/pY5c+fhcrkYHj4QBserpqrWYMmyjN1uTzlswjwAAvTBkmXZ4kszeXDN3ZNJZizJssULpqoKiqJa90y+brIl6odZqNazkyTJotqSJIlEInEUI5Iuq/4YNO2DklVFVRQr4dS8h8lfOyTr0OknkiSw2eyWrIqSIB4fJqskYTtNxoCzWtEy/H4+9vGbGTd+vEX8ZsIkQ3780b+wauVKEMabrmpMmFjFnXd/Dq/PR2trCw8/9CBtra16Zi96Vu8VV17NVddcg6ZpvPfuO7y8dCnRWMTY5QoCgQB33P05ysrKCYdDPPbnP1OzayeqMRMJBLNnz+G22+9AkiT27K7hqScep6enx8g6BqfDwadu+wyzZs8B4JWlL7FixVuoqopJtlxUVMyX/+Ueg2m7jb8+9hdqaw8gSxIIgZJQuOyKK7jhxo8BsH7dOl564XkGB/sNBnEJp9PJF7/8FUrLygmFQrzw3LNsXL8eDV0ZEwmF6uppfP4LX0SSJGoPHODpJx+nra0VSZLQACWucOfddzNn3nxUVeXtFW+xbOlSFEXRZVUUsrICfOu7/35az/KsVjRN04hEozTU11NdPQ3N6bTmM824np2TQ2l5OdZSpGnkFxRYDNIOu4OiwmLcLrc140mShD/Tb/WTnpFBaVkp8VjMWO4E6RkZFm2mJMnk5ecTDAWHiOgQ5OTmWsrv9ngoKh6L35+pvwWaht3uwOv1Wv1kZgUoLStHU1WEAFWD3NxcS1a7w05BwRhURUFIEgJdGbMMfl0An89HSUkJoVBQbyMEDocDp9MF6DNkTk4OpeVlQ6yYikpeQb51D7fbTWFxMW63B2HE/VRFxesbCuv4M/yUlpXpsgiBqmqkGwyap4OzlvFR0zRampt58om/8pc/PcK/fuOb3HDjx/CkpXHkSAc//uEPeOapJ/mfX/3GOFtAxlxWZVlClm3WfZKXGsBaHmRZtpYPnT9/aKkx20iSlEI1n8x4aF5PXjr1g7z0fiRJIEmydXbB8GVRb6MvrycjK5qGmiTLBymrubwOLZ3KUcu8Kevumhq+/Y2v09TUxI/vv58LL74Eh8Nx3Od5Vs9omzZu4Plnn2HJhRexfds2Lr/iqqM40hRFwSbL2O2O47IejoTkayO9qcPvYc48I7U51gEPyX2bdtKJ+jmWrAiBfIZklSRpROX5SG4GFEWheGwJ3/63fyfD7z/qiwsh2LZ1K3aHHZsxg+Xl5TNn7lxkm40jR46wY/s2BgcGjOVM/8zUadMoKSlFCMGB/fvYu2ePPkMYbTweD4vPvwCXy0U4HGLH9u2WjWca02Xl5UybPgNVVWlrbWXXzp36cmYqixAsWLCQ7Jwc4vE4+/ft5fChQ9ZMpGkagUCAhYvPQ5Zluru72LF9O729PfrnNf37T585k9LSMgRQX1/Hzh079NnIsAPtNhsXX3opTqeLSDjM7poaGhrqU2bRktJSZsycBcCRIx3s2LaNwcFBS1ZN01i4eLF+cFkiwaGDtezbt0/fYBmy+jMzOe/8C0576TyrFc18M51Op7HjJGnJ0Nvs27uHaDQCBkHwxEmTmTFzJrLNRk93FxvWraOjvQ3QrTjJOJGtuHgskiRRX1fHynffIZFImKYVgewAs+fOxel0Eg5H2LljOzt37NDTzoUgFo9z3nkX6HYjGh0d7axds5qurk5kY3kSksT48RMIZGcTj8U4sH8/q1e9b/WjahrlFRXMm78ASZLo6+1l08YNNDU0WJuWeDxOTk4uY8eWIGSZluZmVr77DvF43LqH2+1m8Xnn43A4iUQi7K7ZxYb16/TlWAjisRjzFixk2vQZCCHo7upi/bq1tLe16UulsdxOrKoikBUgHotx+NBh3ksaE0VVKSktZcHCRafNU3dWK5qOkU1I3TWmcdkVV3DxJZdZb7AnLc3aghcVFfPpz35Wd38k2U2B7BzrzZw1ew5l5RUpfiOHw4HXqxvG6enpXHPd9Vx40cXoPi+93/QMv7GMCCorx3HHXXcTj8ewUmckQV5ePkIInC4X551/gTUDgv6iuN0eZMM+yy8o4JOfupVoNJrylXNycqzvNmnKVPLy8y1bEkC2yaSlpSGEwOvzcdmVVzJ/4SJjttJl9fmGjPjisSV85o47iUWjQ7IKyMsvACFwOJ3MnTeP8RMmpNiKLrd7RFb0k8U5oGjHR05OrnUY2HAGa09aGmPT0kbMLjDbZGZm4s/MPCYtus1mIz+/ABghpmc4ZNO8Xv0wr2Psq2RZJjsnh0B29tGU7EY/LpeboqLiEfsR+n+SkZFBenr6Mfux2+3k5uaRm5t3TFp8j8eDxzP2mN9XlvUTaDKzso5LFX+qOOcVrb29jZaW5pRZwOPxkG3MBPF4nM4jR/SZAkDog5WR4SczMxNVUxkcGKS3txdFSWC+5XaHnazMLNweD/F4nJ7ubsLhsKW0kiSR5k0jMzMLWZb1MzN7eqyTgwFkm42srCzrQIuenh76+/pSHqDL5SI3NxdJlonH43R3dxEKhfRDx3TXP+np6fqDV1WCwSC9vT3G6XTCeBlkAoFsXG43iUSCvt5egsFgys7Uk5ZGVlYWNlkmFArR09NNPD4kqyRJBAIBPGl6ompfXx+9Pd1J2VcaDoeTvPwhN8mp4JxVNHMb/tCDv+Pxxx5NCZ3Mmj2H//jhfWT4/TQ3NvJv3/4mdYcP6z4jTVeAO+68izs/9w9oGrz6yjL+8PvfEQ6FDP8V5BeM4fs//BGTp0wlGBzkvx64n3VrVg8tWwIuuexyvvPd7yHLMtu2buFnD9xPR0eHZWC73C6+8a3vcMmllwPw5F8f48kn/mo5QTVVpXLceH7xq9+QkZFBS3Mz9//4h2zfts04oU+QSMS57fY7+cKXvoymabz79gp+++tf0dPTre8ehYTb4+a/f/FLJk2ZSnBwkN/95le8sXw5qmY4bOMJFp93Pvf/189ACGpqdvHz/3qAusOH9VPsNN0e/P4Pf8TlV16FpmksW/oSDz34OxKJuLUBysvL5y+PP8HpZDWc1X60kWoGVFW1/GjPPv0k3/n373PNdTekJD66XC7rgK5EIkFn5xFisRgWw4eAjIwMMjL8aJrG4MAAfX19KKpitbHZbAQCAZwul344V1dXSpqSOUtkZmYiSZIxS/SgJPmeJEkic9iMNtDfn/I9HQ4HObm5lqxmP8lLVHp6Ohl+3cGsz2j67GvKKskS2dk5+sFoiQS9vb3WOeom3B4PgUAASZIIh8P09vQcdSa6OaMJIejr66Ovr3fID46G3WYnLz+fvXv2fLT8aCeCpuk2Wl5eXspuyHIxaPrhpvn5BUcda5MMr8+H1+dL8vqTEqyWJYnsnByjz9QjaqzIgNuN2+3WK8aO0Y/f7zdOihtCsu/MZrORm5dqXyXLIoQgLS2NtLS0kWXVNH25DgSsk/1GktXlcpFfUHBcWdPT0/H5UgtATvfUFDjHFQ30Gc40qI8aBHOAObYRO5KTd4RGVujrhPdJajtSm+M9qOEK8bdq80HIeqo45xUtFArR29en+7gkCYfDjsftQZJlYrEY4VBIz1QAMKqm3B4PDofDWkbC4VBKyYFNlvH6fEiShKqq+uly0WjKLOdyOnXD2TizKhwO6yEbgXV6sMfoR1VVIpEIsWjUyKjQEAjsDgcej8fatITDYZREwpLVXJ5NWSORCKFgMMlC0pAk2YpBqqpKOBwmGomYUx0CcDgdeDz6kmjJGo9biiYkQZonDbvDgaapRCJRotHIUEgOsNntpKWlfTQdtseDOUu99cbrNDTUG/nFgnHjxvPxmz+BQ5ZpamzktVeX0XnkCKA3kWWJK6++xvKUb964gXffMZyThrc9MyuTO+76HD6fj4H+fl59dRn79uwxBlk30OfOm8fV116PpmnU1h7gjeXL6e7uRpb1jApJEtzy6dsoL68gGo3y/nsr2bRhg7Gz1ZfgsSUl3PbZO4yjEFt57dVXaGluNjYTem7+tdfdwKw5c9A0jZpdO1n+6qu6w9bwkTldLr745a/i9XoZHBxgxZtvsnXLFiv2m4jHmTFrJtff+DE0TaOxvp5XX1nGkSMdRmxTPw78ts/cTml5OdFojA3r17HqvZVWXFZVVQqLirjjrs/9/SmamYqXmZVFcXGxNSNlZ2dbcT6X201+fgEulwud9UOflNLShsrDfOkZFBUX6ztBsM6nNI8Dkm02srNzCI0NIgkJhJ4zlpmVlWSfecgvKMDr8xptDJ+V22P5pjL9mRQWFVlhHVVVycnJtQLZTqeLvLx8PQhvRAYSiThe35CsXq+PoqIi46XQd7Y2h93yIdpk3Z0ytmQssqSPQSKRICuQbS2FLrebvPx8XG63lYqkqSouz5CsGRkZhqz6oCqqSo5xpufp4pxVNNDf6IULF3HtDTdaD0yWJH3LDuTl5XHNddfr/qSkTNLk3LbJU6YwYcIEa7kCjCXYgQC8Xi9LLrxoKOsiyUcG+s5ybEkJBWPG6EW1SRmt5k7M4XAwc/Zspk2fnmLoS0nB7ezsbK646upUWSHlANaKykrGlpQMyWEsj2Y6k9vjYeHi85i3YAFWMNSQ1bxHwZgCrrvhRkvhk2UVxthMmVrNxKpJKXKYmR9/l5sBDXA4ndhsNivVRqDPFuag2O32lJQZE2YbSZIM20RL2aVpmp42aEYHklOKktuA/hDMjN7kfqy0bEMO044yMZKsmqamhJgQwmpjvkypcqTuhE8kqxDHltVMyLTZbMeV9XRwTiuaEILt27fhS0+3bDQ0jUB2NlOrp+FwONckBgQAAB1GSURBVAgGB9m+dRuDgwNJbgKJiopyKirH6XZLYwMHDxwwjHndQHZ73EydOg1/ZiaxWIyaml10dhwZemAS5OePYcrUqUiSRFdXJ3t27yYSDltyyDYbkydP0V0JwMHaWurqDulLktHGl57O7DlzsdvthINBdu3aSU9PN0JIQ8H38nLGjZ8AmkZrawsH9u8nGo3qsxS6W2TGrNn4/X4jU2Qfba0t+sZD082B3Nxcps+YaRz93cPePbsJDg5acmhoTK2ezpgxYwD9CMvaA/uNrBa9jdvtZr5xtvyp4pxVNGGEkla8+QabN21E0rd7qKrK9Bkzqawch91up6uzi0f//AiNDQ1WZECSZW659dOWom3dvJlH//wnIpEIkpFxmpuXx9e/+R38mZlEIhGWvvgCmzZssJYTDbhgyYVMmjwZSZI4fOgQD//+Qbo6OxGSPoM4nS6+8i/3kJevB9dXvb+SF557NiWVu6y8gsmTp2BPT6e7u4tnnnySmpqdVmQgnojziU9+inHjJ6ChUbNzJ488/Af6+nqtGc7ldvPD++7H7/cTCYd58/XXeGfFCv2lEBCPJ5g3fwHV06YjSRKNDQ385U+P0NhQbyRUQiwe4+vf+g4FxkuxaeMGHv3zn3THsBEZyMnJZcas2af3vEaKDKTaGmbMTYeqKsZbrU/3fytG7ZOJDDzz1JPc/58/47rrb9AdtkmOT9M+0jSNWCxmxTFhaIkxXQ+qqhCPJ1BVBTN+aGaT2mw2K9s0OQvXXMrMP/F43IpzmhVZsiRjM5ZMSZKIxWJ6ZXfS97TJMnbDPjKP904k4kfJai53qqoambqJFFnNAhbzenIWrmRsakxZE4kEcaPC3JRVkmRredfdLTFisdTIgSzLOBwO9uze/QFEBjQNdbCN5n4NOS2b/HQHsjD3YwnCve0013cQEl7yJ5ST45KN6x8OXC4Xbo9n5MiA8bPT6UTTjk6aBCz7SJZtx4wemNmmx8qIACylHMmrb7az2+2WLXksWR0Ox1E5X2Yb849ugzlSrpv/ypKE5HCkeP2HyyoblK7DZ5jU72O30uFHkvVUkXInTVWI9zWz/92/8PAePxMW3cTdiwv0KmM1QaT7EHtXv8OyN3fTp6ZTddOdXLOwmBy3/UM7pqexoYGtWzZbCpOR4aeishJZlunr66Ohvo5wOGy1F0JQUlJKdk4OQghaWlpoaW5KMXydThdVVVXY7HZi0Sj1DfX0dHcbNpFA1XQWnZLSUgB6urtpbGywYpTmn3HjxpOekUE8Hqe1pYWOjvaUfny+dMZPmIAkSQwODtBQX08wGLSua5pGaWkZObm5gJ4d21Bfn3IPm2xjSnU1druNaDRGU2MjnZ1HhmRVVbJzcigrLwcwxqSecDiUojjjx0/Al56Ooih0tLfT2toy1I+mkebzMXFi1Qe0GVBihBvW8/rrb/FGx0yyqmOGQSlQlR7qd25i/cZ+8uYuYZ79MK/+9nHyiz7P4vE5+AV/kyX0WDAdtu++s4KDB2ut3d3kyVMYW/J5ZFmmtaWZ5599hra2NuNT+hLxyVtuZaERYN65fRvLXl5qLVeappGdncPX/vXr+P1+BgYHeev119m+fSuSJFve9QuWXMjYkhI0TaPu8GGee/ZpOjuPWMUokiTxz1/8Mr70dKLRKOvXreW9le8O9WNkbpRX3IPD4aCjvYOXXnyBhvp64+Hry+htn7mdgOEb3L93L08/9aRVraUZBvr3fvBD0tMzCAaDrHz3HdatWW24NCAWi7Nw0WJKSvXU9ZbmJp595inaWluQZZvls/vyV+9hvM9HLBZj69YtvP7qqyQM57KqKJSWl1N577gPKMNWtuMsqGbhzIms2ZGWapv1t3GwpZWGQBUfX7yASlcuB5/+Jjv3fZzKggD+dIESDzM40M9gWEGVHHh9GfjcMko0yGBUwy4iRBIywu7BLSKEI1E0lx9fmhuXnGoLngjmtD9v/kLmzZ+PZBjPgeyAtTzl5uVx+ZVXpcwSQgjKKyutN3NCVRUOpzNllnC73XiMIpi0tDQWnXc+4ydONGYAvTStsKjYmhEKi4u56pprCYfDKTOauQlwOBxMnzGD7JyclFnCn5lpyRrIzuaSSy9jYGBgaMxVlYpx4yxZyyoquOljH0ex7qEv2W63Ww9XeTzMmz+fsWPHWulOiqoyZswY6x55+flcedXVBINJ9Q2aRl5eniXr5MmTSUtKGNWMpMsPLMNWSDYc2ZVUlRdQUC9ITkyN9/XoyXLpPrxuN3ZXMVVVAzzfdoSevgo0T4Jo4waWPreUFduaCbkmsujam7nrimza3nuJP273MiNzO5vqJdwVF3NpRg1vvLeFvimf445rFlCdbz9lW0/TNMaNH8/5S5Zgtx9ts2RlBZi/YGGqTyn1L0pLyygpKbXul/x50JUu2dE6UkZEXl4eucbyNlIbh8PBuPETqBw3fkTbSQiB3+9n7rz5I8g61E9hYRFjxhQeU1aXy8XkKVOZNHnKMWUNBLKZvyBwTFntdjtl5RV6/ekINukH6rBVVA1VS10KNUXBZRMEMnw47DaEcOBwycQSMZRElFDnPlYu3U4T07jxywtofP51Dq/4K3+R59Bx/29Z6a5EvfFaJhcdYtN7T/DsuAVcet4Ulj29mk2FORRfMZEAp76D1VSNaDSGoqjWZ+WkXRroSX1miMn8K9l4V1UVJZFASc4ESdpVJtMQJGfYmteT6yA1TbOUSTYpEYyl1KJESHqAsrHrBP2hm21MWQUGJULSrtOsyzQVUTqhrPpmx0w8GElWc5ctpcgaT8lxlAyqidPByZ/XqWnIsozb5dQNb01ClvS9qKaGGWzaQR0Oxsy+iPOq/SiBSlp6EyQ8KnUXzmLfJj/VC+YwoVvhwHt9ZE6pZtqsDlY+tJre7n4ip5h/KdAH50+PPMybbyy33gpN1aiaPJnP//MX8Hl9tLS08Jtf/S+trc3W2yhLMtdcd70eitE03n7rTV54/tkkJ6ggkJ3NP/7TF6iorCQUCvLIHx5ix/btJGVwMXfefO7+h39EkiR27dzBo3/+Mz09XZarwuFwccdddzJ33gIAXnj+Od5Y/prBE6LLWjx2LPd+45t4vT7aWlv50x8f5sD+fUiyXqmeUBJcdfW13PzJWwBYs2oVzz7zFAMDOiWCJAQOp5uv3Xsv5RWVBINBnn7yCdatWaO7qYzY7PQZM/nKV+9BEoL9+/by2F/+Qmtrs7WkJuIJ/vGfv8CChYtQVZU331jOi889R8Lyo6kEAgH+40c/PqXnZGJERbPyt5KmF6fThWR30RuMEosnUOx99LTHSC/KwGVXiPd102Nzk+5Lw21z46ysxq+oJGJdZCyu5v19gpICD/ZeF7JtDCVZGXhtHSAJpNPwxWkYu7KyMqqnTbfk1jSNouJibDbdr+b2uJkydSqFhYW6AqDPNAXGEgS63TJz5mwjK8IMtqeT5tXz5202G5XjxmN3OFNCPxUVFVZ7vz+TadOn6/WShhx2u90IaOv9FBcVM3v2HFRNs7z+2TnZlp3mdrupmlSVwsmrqirFxcWWrNk5OUyfMZNwKDQUunLYSTOoF2w2G2Vl5ZafTLfTFMrKyq1l2OdL18ekqMjaDKiKQiA72/hugvz8McycPQfVyDpWNRWfz3eUe+ZkkfopTQNVn3bjikZCUYkrKg5ZQmTl4PP60Pbto2tSDllptWw+MoOpFfkUZDlwZOZg76yled9+2ssmk+Poo7svwZHeOJ5YHEUVJOIKmqqganHimpJS3q8fwZiS8nVCaJrGJZdexpVXX2MYqliefzPTwO/P5OZPfHLICE/yn5lKM2VqNZMmT7GSKE3fl9nG6XRx2RVXojMF6WEdczk0Fa147Fg+9enbrOUIsJZf842dO28es+bMMaU35BiSNcPv59rrb0Q1A/hJsuq/CiZMnMi48eNTguKmrKDbaRdceCHnL1liOd6F0NmGTFnHFBZy8ydvsZy15vcxIw1CCKZNn86UqVNTZBVC+qAULYHWd5iNe5po2NPHQPo2aqZmMac0A8mTz4TSctprNrHi5SPUOLvpmHIHN0/MosBjI1Eyg3ljmli+9VWeHNxPsdZDMG0Mcm4e+du3cSScRUNjO93799LW00ZWX5C+lnb6YodoPdjAwp5pFGTLR3HWngiJRAKb4RVPNnrN5cscZDOjw7ye/C8cTVcw3PDVH2SqDym5jSQEYlgaTfJ1TdMQkoRNOvY9zH6G+6qGfx9ZluGEsgKM3GakLIyjZBXiuM7lU8VR6inCHRyxlzOltBXhGKCuK8rcMkB4KZg2lwWSQu8LG6gP5nPdP8ynONONLIGcOY55V11Kn/w26/dtZ2ssj6oLilgww83+zQGmXlCIOy4jpHTGz7OTnZaOGs+k+tJy9nscxHpiSDmndvCAEIItmzelcEnkFxQwf/4CgxKhg21bt+oFIcZsJ4Rg+oyZlJaVIYRg/7597N5do88iRhuPJ40lF12Iy+UmFAqxfetWWlqajXCbHvcrr6hk5qxZqKpKS0szO7ZvJxQMWm2EECxafB45ubnE43H27tnNwdraFJdBdnY2512wBFmW6erqYvvWLXT3dBsuB50SYebs2ZSXVyCAusOH2b5tq0GJoLslbDYbl19xJU6Xi3A4zK6dO6ivq0uhRCgrL7eoszo62tm6ZQuDAwOWrJqmcd4FF5CTk0siHqe29sAQTYQha2ZWFksuvOgDcthKNqQx8/nE3bP4mKKCbMNmM8lTBMKezdjqK7hzwsUoyDjcTmymfSW7cRfO5MrbpnBJQkVDJ22z2wUV9/yYy4SELNthfjWqBpLNiU2U8w/TryKBHYfDfkrLptk2+eEhBBMnVjF7js690dvby7YtW+joaLcGTJJl8gsKKCkpAUmiqamRdWtWDyUTqhpZgQDzFy7E6XQRjUTYs2c3u3busJYW0/6ZMXMGmgZdnZ1s2bRJp0SQh9iDJk2eYnBvxDh86BDr1q7Rq6SE7rEvL69g4aLFSJJEf38f27dvo6mx0Zq9EokEYwoLKSstQ5Nl2tvbWLd2jWVLapqGy+ViyUUX43A6iUajHNi/n40b1luzUTweIx6PM3PmLDQh6O3pYeuWzXS0tw+lHQFTq6vJzs4mnkjQ2NCQIquiKJSUlOjUCx/MrlPop6S4bNbvqQ9fINkcuGQ7mhjBgJdsOJwydqf53I2p2p1m/a5ptiRFcSLZHDiP6ufE0E0hnRLh0suu0HdpQuBxe6xBLi4q5rbP3q4rhmkTCb0EThhv5uw5cxg3Ts/iMPeTdrvdqgDypadz7fU3cPEll1ruBg0Nn9dnfb/KceO583OfIx6L622EQAjJoEIVuJwuzrtgCTOMGdAYWVwul5VAWZBfwC2f+rROZ5/kUwhkBSwetMlTpjJmTKFVrwn6km9SIvh8Pi6/4koWLT5vSFZN0w+cMBy4xWNLuP2Ou4jFYkmyCrKzcwDdYTt33nwmVk1K6cfldFl8HqeDkXedJ7rZSEp2nGvJ9xvJNvi/IDMzi6Li4qMYIUHPOB1OczUcGRl+MjL8x7wuSVKKM3Y4hMAqgTsWzIr1ZEK94XC53RS43ce8DjoJ3/ASuGTYbDayc3Ks0sCRoFMiHHtMdJLCzA/87K2zkpX7VHCW1j+PYhjO2cRHE7FoTM83M2wsGObB1jRi8fiQOwBApEYGhjzlqhWbEIan3PK2D2drFEMkxsKwY1JPEdHASJs2bTYzZ00MtUA2UslN6FGMoSp0SI10DOWbDVXVCyGw2e2WrGY+WkrOvzEmlqyJuFV8MpKsZs5asqzHIuc7GZyzimb6jn7x3//Fww89aDmXVVVl1uw5/Pv3f0CG3099XR3f+7dvU19XZ/G1yrKNz9x+B3fcdTeqqvL8s8/wxz/8nnAobO3E8vPz+e73f8CkKVPo7+/j/h/fx/q1a5IUSXDxJZfwze98F7vdzrq1a/j5f/0nR44kcW+4XNz7jW9x8SWXIoTg4d8/yNNPPYGq6B571cjg+O//+SXp6Rk0NtTzwE9+zI4d2y1DPZGIc+ttt/NPX/giqqry2ivL+P3vfktvTw+SrLtC3C43D/zs57qsfX385tf/y5uvv6770dCVd9Hi87nv/p9is9nYumUz//PfP9N3p7Jktfnu937AZZdfjiYETz3xOH986PcklASSoZy5efn88c+PcjrcG+esoplv900fv5klF16UYkBnZmZaJMW5eXncc+/XCQYHgSHjV6eI0j+z+LzzKSkttbgoJEnC6XJRWFwE6OV0t99xF9ddf0NS/FAnJTZn0QkTq/jmd/6NaDRqKaPdbqe8osKS64qrrqZ62nSLlh00vD4fHo9u3+Xk5vFPX/yyxfpoOViTohiz584jNy/fCJeZMUo7RUb0wO3xcPMnPsmFF15kZXkIBFnZASO8BhUVldxz778SDodTgurjxo23HMAXLLmQ0jKTcFn3+7hcblwnsCOPhXNW0UxMrKpi9ty5R6WwmD97PB6mTZ8+VLySAn1WLBgzxuKiYNg99Jilg4lVVUOedOOzyQ7OQCCQauwb90qWqaS0dKhcLglmBMLtdjN5ypSUUrrh7axMkWFtkjNFKseNp6JynCmILqveCAT4MzOZ7p95TFmFEBQWFTGmcEjBk2UYLtfJ4JxXtL6+Xrq7u6zCW4fTidfrRZIkotEowcFBEoqSkqXg9XqtWshQKKTTDCTZMzabjQy/zuioqiqDgwNEItGU4XW53fgMYphYLEYwGNQpnnSfAZLQGRidRq5bKBTS2Ygsnx84HE7rHvF4jMHBoJXdYZZs+Lw+nC4XQgjC4RADA4PDbC+JzMwsS9ZgcFDPi0uS1uVy4fX5EEA0GiMYHDTsL2EpmM/ns9LVdZqIcIpd63A48B2HBPBEOGcVzQz8v7F8OYcPHbImo3HjJ/DJT92KbFAiLFv6EkcMSgQzw/aa666zPOWbNm7gnRVvGVTnAk3Tef0/9/l/wudLp7+/j2VLl7Jnd42RYQvxhF5VdL2R/VF7YD/LX32F7u5uI8NWf3i3ffYOKioriUYivL/yXdavX2fQWun9lJSWccdddyNJEq2trbyydCnNzWaWiUY8nuDGm25ijpGntmvnTl5Z9jLxmOmwVXG5XHz1nv+H1+djYGCAN5e/xpYtm5FlPfYbj8eZNXsON338ZjRNo6G+nmVLX6Sjo0PPsDUqtm6/827KK3T6hg3r1rLy3XetAhhVVSgqHsvn/vHzf391neZ7lZefr9c8Gv87prDQilmmpaVRWlZGZlYmJiUCAmu2AsgKBBg3bryVi6ai4U3zWsfV2G12CgsLUVUliRJBsbJnQfdvlZaVk5Oj85yZy695Mq/NZiMnN5dx48ahqkbmhqqSm5dnGf0et4expaWkeb0plAj+zEyrH7/fr8tq7EpVI0PE3E3a7XbyCwqYMLHKoDvQS+3yCwosmy/Na45JwKJEUFWVtCRZA9k5VBpObD37Q02hmjgdnLOKBrptNG/+Am648aYUigJzQHLz8rjqmmvRVDVln5Rsz02aNJkJEyamZL6aD02g89Oef8GSobif0cbsQ5IkiseWUDCm8KgKKJupAA4HM2fNYtr0GSltkgPogexsLr/iyhFkHcqoqKiopKS0bERZQbdHFyxazLz5C1JkNV0WevpPAddcN7SpOUpWu50pU6dSNWnSMFnF3y8lAgxxRiTv0oZoBnSFUIVIGTRNVVFJLrVLpRAAPaBtXpdMmvRhSqC7D459j2QOWb10LfVUlFRZ9dlEPyxMxTS4zd21SSkvY1ItDLVJkdX4N1lWMawfc2ySza1UWeWU/0sek9Od1c5pRRNCsGnj+pR0Fk3Ty/9nz5mLw+lkcHCAjevX0z/Qb5nHkiQxfsJEayd5+NAh9uzZnZRCLUjzeJg1Zw5ZWQG9MmjzZtrb21ISHwuLipk5azaSJNHR0c72bdt0SgQDsmxj+swZFtv23t27qa09kFR/ABkZ6SxctBib3U4oFGTb1q16tbvQDXVNVRk3fgKTp0zBpG/YXbOLaCRqvVw2m435CxeRlZVFPBajZtdOGhsbrX5UTaOgoMA606Crq5Md27YxMDCQ8nLOnjuX4mKdsftgbS17dtdYSZqapuHxeLjw4ktO61mds4omjIDw6lWr2Ld3n7XLUjWV6mnTmTy1GofDQW9PDy8+/xyNjQ1IQkJDQ7bJ3PyJWyxFq6nZxROPPapTIhh2WG6uTiuflRUgGomw4q032Lx5k35SGICAxedfYPFZNDU28uxTT9LVaaRyo+F0OknPSKewsEh/KTZtZOmLL6RQIpSWljJ5ylR6+3o5VFvLK8uW0tzYhEiiyLrp45/Q3R7Agf37efKvf6Wvrw9ZkhGSwOV2UTluPFlZWUSiUd5/7z1WrnwHTdUJ/+KJOHPmzWPO3Hn6xqOlhReef46mxkbrjKlYPE5mZqb1UuzauYPHH3sUJaEgJN1hm52Tw8LF553e8xqJEuHDhrlcvPTC8zzz9JP8729+R2ZmluVvSqZEeOBnP+ea665PKZoYTokQj8V0FwemnT6cEkElkYijKElnYA6jRFCUBIqSWpxiMxIqTUoEJZEwMoX1ITVDVGZRSCwWS3FfaKpKNBZj9fvv8Z8P/JRYLM49X/saF19yKW6PW1/iECOGoBTr+whLFtmgRBhJ1mT6hqMpEYYYkUxZ4/G4fkCHZnri9DDWB0eJcBZA0zS6OjtpbmpicGCQXTt34vf7mVhVhcPhTGnrcDhwu93HpURwOJ3YRygdg6PttOHXzTaS5MBuO5pc2Px3REqEYW2SKRE0TSMSibD6/fe474f/QXtbG6qq8rvf/JrKceOonjZ9iIoryQY9nqzmNV1pOKqN2c5UuOPJasZ5h4+Z/ufUs27OSkUDqKs7zIYN6zlYW8v3v/tvuD1p/P7hP+q0Skk601Bfz5ZNmwy3wocn7+mgs6ODB+7/Ce1tbdZs1dPTTc2uncRisf+TO+FvB0FjQz2Dg3pI72SXw7NS0cyt/JVXXUVjQwMH9u+jakq1FSTWGMq4eOKvj/Haa6+lbqHOdgg9cB8c7OfwoYNGlb3+0Hp7enjwt7/F7fZY6dpnDYxYaTQapbHhMFlZgdSsmOPgrFQ00LlpL7/iKtxuD3/8w+/5xre+Q3a2HsS2yboDtLyiAknIhJIoD84tCCoNgr2h/9JrKEOh0Icn1gmhkZ9fQF5+vu5gPolowVm5GTBhxt0OHaxl4sQqyykbj8dpqK/niHEczrm2ZFrQjALspNQjc7MyQkz97IEht8vpoqS0FL9xeszxcFYrGnDUzsj8v+SS/3MTev2noijW4V+aEDovmc2GdA68P5IQVj3DiTYHZ72ifTSh6dGJeIim9Y/w00fW0d0fxpFZyISr/oHbLppCUbqELJ3tqnbyGFW0Mw4NNJV4fzP7X3+Uv77ehWNeFUXOBEpvPXt2KeRf+CnuuHoKuekOPiq6dtZuBj6y0ECN99G67W1eeGEbg7O+wFc+s5AiSUPt2sOa4O/5zSsvUVaZy1UzC/GeIm/c2YpzvgrqXIOGhtK9k91rV7OmYxJX37mAYodddzznljH9yvmUdq1n5eZD1PUrSWnf5zZGFe1MQ4NE1xE6O7uJ5JeR49appyQhEDYPttxSphUk6G3vom9AOY0ykLMTo4p2xqFvBDQhkFwunGLIpQESkmzH6dYPAjunN9XDMKpoZxpCYEvPwOeRURtrqOs1AvFoaFocNdpHS5uM1+vD65HPehfHyWJU0c4wBCDnTKB08niKo++zakUD3XEVVVFQgkdo2rqbOrmSGVPHUuA//YzWsw2ju84zDSGQnAWUzbyUK2at4+WXn+TNsiuZ5IdYyzbWvHkA2/wbuXBaMdn2j8aOE0b9aB8ONA0lESPccYANj3+Pnyxroj8YwZtdzLxb7+WzV8ynPMuBw/bRWXBGFe1DgqapaEqCRCxMKKYa5YICm8uD027HJv3fmZbOJowq2lmAYyVcfpQwaqOdBfioKlcyPjpGwCjOaowq2ijOCEYVbRRnBKOKNoozglFFG8UZwf8HiHc3M6J2FywAAAAASUVORK5CYII=)
Consider the situation shown in figure. Water (mw = 4/3) is filled in a beaker upto a height of 10 cm. A plane mirror is fixed at a height of 5 cm from the surface of water. Distance of image from the mirror after reflection from the mirror after reflection from it of an object O at the bottom of the beaker is
![](data:image/png;base64,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)
physics-General
physics-
One side of a glass slab is silvered as shown in figure. A ray of light is incident on the other side at angle of incidence i = 45º. Refractive index of glass is given as 1.5. The deviation of the ray of light from its initial path when it comes out of the slab is
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)
One side of a glass slab is silvered as shown in figure. A ray of light is incident on the other side at angle of incidence i = 45º. Refractive index of glass is given as 1.5. The deviation of the ray of light from its initial path when it comes out of the slab is
![](data:image/png;base64,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)
physics-General
Physics-
On a hypothetical scale X, the ice point is 40° and the steam point is 120° For another scale Y the ice point and steam points are 30°
On a hypothetical scale X, the ice point is 40° and the steam point is 120° For another scale Y the ice point and steam points are 30°
Physics-General
Physics-
Express 0K on Fahrenheit scale.
Express 0K on Fahrenheit scale.
Physics-General
physics-
A uniform ladder of length 5 m is placed against the wall as shown in the figure. If coefficient of friction
is the same for both the walls, what is the minimum value of m for it not to slip?
![](data:image/png;base64,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)
A uniform ladder of length 5 m is placed against the wall as shown in the figure. If coefficient of friction
is the same for both the walls, what is the minimum value of m for it not to slip?
![](data:image/png;base64,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)
physics-General
physics-
Two light vertical springs with equal natural lengths and spring constants
and
are separated by a distance l. Their upper ends are fixed to the ceiling and their lower ends to the ends A and B of a light horizontal rod AB. A vertical downward force F is applied at point C on the rod. AB will remain horizontal in equilibrium if the distance AC is
![](data:image/png;base64,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)
Two light vertical springs with equal natural lengths and spring constants
and
are separated by a distance l. Their upper ends are fixed to the ceiling and their lower ends to the ends A and B of a light horizontal rod AB. A vertical downward force F is applied at point C on the rod. AB will remain horizontal in equilibrium if the distance AC is
![](data:image/png;base64,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)
physics-General
Physics-
The spool shown in fig is placed on a rough horizontal surface and has inner radius
and outer radius R. The angle
between the applied force and the horizontal can be varied. The critical angle
for which the spool does not roll and remains stationary is given by:
![](data:image/png;base64,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)
The spool shown in fig is placed on a rough horizontal surface and has inner radius
and outer radius R. The angle
between the applied force and the horizontal can be varied. The critical angle
for which the spool does not roll and remains stationary is given by:
![](data:image/png;base64,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)
Physics-General
physics-
Find minimum height of the obstacle so that the sphere can stay in equilibrium.
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)
Find minimum height of the obstacle so that the sphere can stay in equilibrium.
![](data:image/png;base64,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)
physics-General
physics-
Two halves of a round homogeneous cylinder are held together by a thread wrapped round the cylinder with two equal weights. The complete cylinder weighs 31.4 kg. The plane of contact of both of its halves is vertical. For equilibrium of both haves of the cylinder, the minimum value of m is
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zYS5cuzRdc4gkI8KbIYEZGhkrvyXG1atWSU0455Yi/wb45di7/73//U4nqnJQB1yVpcMuWLet49fS1a9eqLUAsXJB2Xt9g3QC1ZMmSKh0UmxL1652Rk9yzJE0mNdR9992nYiaIlcBOLVq06BE2tW6YKky8eCDcyg9hwHVJZBnkIgMMmWu+/fZb9frGZbRx40bZsmWLik/YtWuXWmHC9cXr+MCBA2pfWrRB5YsXL1bZzxltddqno22Az1I09iupUt3e4IkMuC6IyYtOJcTrlFGXPK9sT2EJl8rq+Dcvu+wy9QrGN8tKWYMGDaRRo0Zqlt6kSRO1ANG8eXOV1YUYVgp4kMqJVzcjeufOnZWti6+3e/fuysPQv39/dZPZv3XTTTepYB3+NueASRHoYw3WmGDheiPDJOeC2w1zgUgugMWTwEOmHzTMD8wLp2XSjLogbEO9GTKejZGSnbPYpvw/CTd4kIIdqxv9HIuZAMDYypgY/BcT4ZxzzlH/ZTSvWLGi+rxy5crqobzooovUA8nDQnmn+vXrK88DD+ENN9ygHiQevg4dOsjtt98unTp1knvuuUetCGKrE4xDtXQiyQjEGTBggAwePFg9NDyQ+hwNuA4J/yUjJReZEY4kINw4KqpzI5nQ1KtXT33ODWZk4yZzw3HiAwqjJMumAAIsuLyY6BG3QJ7d0047TdmzjKIABqDB7NFEaFxD3kb63wZcB8VowkXmFa1zh2G3Yr/yeuU1y+sW+xYXE6CzbQf7FzsYm/jrr79WwSr4XvkdeCc+/fRTtZ19/vz5MmfOHFVulaqLjGSTJ0+WiRMnqoze48ePVwVARo0apdxjw4YNU6/8F1980U40nZqaqv6tYyJ4WLp166bMD2x0zBHMEo5r06aNGjG1+cHxPDxsveHhY9LHZI8Hj9GXh46RmXPTbjsmhQTbMKIH+n9DNcDlgdf/NuA6qEB3GKZDXlq+fLkaLTmWGX6k6tixo/pZbOVwxYMxb948O3cZtjK/A68E9RYQDxm2KwsaPGjYtjxsNODmeCaeTAZ37typfg4bGPueh++HH36Q7777Ti1L6+06vBFYGmZrEq48HkauDauCekTlerEPD/MAQPENE+uA6UA/zYDroML14zIq6tEn0mzlwMIEj59lFAWuSIW/FjOF34GNircjPxGXoDPxYKaEOh5pDwubQnmDBBNmFMcwcaVqek6ZyZlLChdcJiD6hvBKj0SYE3r5F5s5GjE66l0XvPYZXfMTCw06Vhi7mxE4lK677jp1PLY45k9OMVLr4Hl+Nw9TThl3mEsKF1xd6hM7bsqUKdan4YlXLZm7+Xl8rNGIV7auTI6tG0qMsPp4QAw14mJyMJJzPA9IsGw1mA7a54ytH8yHbcB1SeGCi1uI4/AWYC9GIuxUfTN5AKIREzxdepRzDiVsVh0JxuQsVEANqaU0lPingy0N8z20nc/1CCYDrksKB1wmPrjJOI5Z+DfffGP1hCe8B/pm4u+MRngh9O9gASOUiHXQx/OwhSrEQlBPmTJl1PH4b4OlEJ00aZLtysOXG0wGXJcUDrjMxnElcRw2XqSBKto+5qZz86ORrnLDxAmXWShhznA8jYeNJer89Mknn6jFDI5nkSGY9DnwPZiEBZMB1yWFAy6xC7r0J+6gSOMTWNblZ/FKkEQuGuHJ4HfgWw3HxmbDI8fTmEiFiv/F/aW3HuXl7nvuuefsc8jrATTguqRwwMV/iS+U43DsRyKA0W4pJmjRbJHhQdHbx7FX87PFtfRkksarP9TDBqz6+GBJQPh5ln3pZ5k8LzvfgOuSwgEXJ7uOHSCkMBKxrBzoloomvy6uLJ36CQ9BfvlptQj44XhaONnU9VuBUXfatGnWp/8I3zOVNzmGa8XiSDAZcF1SOOCyKqRvBjGxkYhCKNqHy6oZy8eRipBKne6TmX8oHy7ib+lzDqcavC5MyESO5eqc4oHTD2CFChXyXEQx4LqkcMBlQyHHYKOyghaJcNrroHFGwVCTpGBikyUZE/kdhFaGquyOR4BoMI6nffjhh1ZPcLFkTFwwxxJBFsxrgomjczAQA5GXDLguKRxwcf1wDH5UZt+RiDV+JjP8PGZGKOiCCZB0bTFe16HEUixmCcfT8ovBQIEjOkE4wZKZkNhP+5Hzs/MNuC4pFLgEuBB5xTG4iyLdGUskmF4IoNxSNOLVrWEgECaUCAjSk0kmhKGySfJgEHHG8UR3BXu4As8hPzvfgOuSQoGLTapLpBKsEmlF9cCgE0IXo1Hg4gOB2qE0Y8YMe8Mk9igRYPkJKHVAfV6jaWC1zfzsfAOuSwoFLhMhRhiO4TUd6fYXTAte8wSVh5vVO6dwx2En86oONnHKKWqQ6R3F+J1D5THDhtYBPCwyBBPnwBuHETy/czDguqRwbFxWznARhRMamFP4P1nj53dHunChxc+lpaWpeNdwbGRe/bjN+F4E9YRTi5gl348++ihPbwF/F9/tggUL8v0eBlyXFA64fhN+Z20qhGMTx1IGXJeUiOAyOusKQUeT6zYaGXBdUiKCyzLz0KFDVXxDsF0KTsqA65ISEdx4yoDrkgy4sZUB1yUZcGMrA65LMuDGVgZcl2TAja0MuC7JgBtbGXBdks4ww0ZBlkqNjk4GXJdEBkIuMvGuThWUTiYZcPMR22FGjhwp/fr1U7mqom1sFdcphRhxKSxCcrlgxx5NYz/Xxx9/HHTLd6LJgJuP9G5TPzViB8h1kOgi06T+zqTw94tcATewYDJB2oTakRrJS41dD3rng27EtHpZRICR5+ztt99W50pEHPXRwmkcSyVNPWegPfbYYyo2OJLfE0ljRCdiLZzot1ByBVzSW+ogEjIM8iXY5++1xu4HHYRO8/qrk1oXOg2TXxocDBo0yPoG0csVcNm1qsElkYWXFfjqBGQvizy5+lz91Jo2bWp9g+jl+ogbqzL7TonJnr7AXgeXHck6fxi7HajxMHDgQDW59FJjAkgeCM1AONkrQ8mAm0N+ApcE1DoHBDkU3KjSE63YR8d2KM6VhIRHKwNuDvkNXAqwcK5kPw+VbyyeIpOO3slswHVAfgWXzOnhZMyJl1gsSi5w//5NduzYLb8f/GfT4V+/7ZGtmzdL5t7g6eYP0r9lq+z4MfLy+AZcZ5Q04P714wZZ+ME46dGpuVzW5AFJ35X14eFdMmfUc9Lu+tpS5fzKUqN+S3lqxGzZrd+Qv2+WD1/vqfqrVq4ql1/TUnqM+FgyI7ifBlxnlATg7pPl04fKHc1qS6ki2QAVKN9GZq9cLiMfbiblzjhTKlWpKMULW30FS0mXN7+QP/askudvqS1FTzxZyp1XXkqcoPtLyN3DF0u4Y2+ygJu5ZoGMHfmOrNRP/aG9snTWu/LKwBdl6OipsmZHjpxpB/dIxsdZ/YMGymvjpsuazMhyqiUBuAdl75Z18sXS2fJgw+x8WoXOvEJuu72TPNF/lKSv3yw7d+6Q1Z8MlevPy87PdUbVhnJnlxRJved5mbd6g2zP3C4blk6SlBrZBexOqHmXLNkeXg6wxAX3b/ll50ZZOOlV6dK6oVQsU1yKFK8jEzYfkp/XTJW7r68mJYpmX88ChU6Usy9vJeOXZWcBykwfL6l1zpfTTs5OJE1/2RqtZFx6puoPR0ll437cw1rFOvkiee79L+VI9PbLpB6N5Vj6C5eW24cslJyF8tMHtpHj6T/1anlrWXh5bhMX3EOyZvZYebxzitQsmw1ooWK15dk3h8o9bZtKy1vvk2f79ZaHUuvJ6dbbrOpNPWXGzDGSUremXNOiozze6xl55M6st96J2f1nNekl3/4WngsuicA9LPP6ZKfSLFChnaRtyx2p9fkbXaUE/aVryZggeeK2vddNitN/7CXyyuLQyZVRMpgKc/o2VQ/8sVkP/PV395Lp6RutHrRTBt9yqRTJ6i9c9HS5rNm9MuaTlfJPdMEfMqXHdXIc16hITRm5Mv90UVpJBe7Cftm1cAuUbyVzN+RONZQx+n4pTX/JmvJ6dlnfI7RtSncpVTCrv1A1GZK2yfo0fyU+uIck/Y075WS+40kXy+CMn6zP/9HXY++XM+k/5lzpMTF37t2ti/pJeXWNzpKnZgZCn7eSCNy/s8DNLvmUJ7hv5g/uVgNuEB2W9LGd5VS+Y7E68tZ3ua/rlqlPSQXMhSJZ1232NuvTf7Rt6StysbpGpeSRKcFT9OeUATdABtzowM0Y11mK8R2LXimjv8k5M+C6ZYF7XFb/cRfIgI9yX7dtGVngcl0Bd7IBN4cMuKFkwHVYBlxnZMB1WNGCu+j55upnCpzXWuZtzA3u0tEPyBn0Z4E7Yrn1YYC2TXvMBvdl41WwZMANW9GBKzLr6WvVzxQo20w+3pzbHZbxRpdsd9jp1WVYkO1hme8/nO0OK1BJ+i8ML8lzUoD7VpeYgfuomZxpHZZfdm6QFXPGyo3nZxcRKVDgBGn13HhZ9tV6+XH/Ydm343tZ8dlk6dLgTKu/oNTvOkyWrFgju/b9Ib/v3SKrMj6Rxxtnx6rSqqX2lnkr18quX/PfuZv44IosywJXeRWywB2zLvebLHN6T6kIuIUvkkGzcj/wmSuGSXUFbgl5ZGp4JlgSgHtQ1i4YL4927CDt2reX1NRUSU1pL+1Tb5X7ew+XFdv+lI3zR0v3e26R9ln9Kao/Rdqn3Cz3PNJf0jbukc3Lp8rTXe6QlHZWf2pWf9axnZ4cKGnrco8wgUqGEfeDntdJIb7jcefJc3NzLyCsGHZH9pusQBl5+K0vrU//0frp3aWM6i8srQalW5/mr6QyFeKhRAZ3S8Y06ftAG7mw1LH2dyxbvYk8+NRQSd+4V3Z9+YH07HaH1CqfXZWdVrJKHen0eD/5aM1e2fP1XBn4xF1Sr0p2xR/aSWddKqndXpQVa3dafyW4DLgOK5HB/TXzO1mycK4s+jxD1VZbvnyZfJY2X+Yv/kIyfzkg+3d/L58umC8LP11i93+etkDmLVoiP+zJ6t+7SZalzZX5aZ9b/csl/bOFMmfhUtm5N//4OwOuw0oGGzceMuA6LAOuMzLgOiwDrjMy4DosA64z8j24EyZMsD71pqgzZsCNvXwJ7rBhw6RgwYLqpOvWrStdu3b1bKtRo4YNrtdzhxlwHRa5cTUMfmp+GnEbNWpkfepN7d69W0qXLq3O1TfgkgqzZs2aKo0nKUa93qh6XqtWLVm3bp31DbwpwD377LMVDFWqVJFRo0appIJebMwddL1i34CLeFVQbJmy+rNnz/Zs4/xomzeHF00WTwEuRVyAwU+tcePG1jeIXq6BaxR77du3T1q3bi3HHHNMUEC81PQchyTa1Cw+Whlwfa5Nmzap/MMkS6ZGhpfbgAEDZOLEieqBO1oZcI18KQOukQ8l8n8FWdKxgwfsbAAAAABJRU5ErkJggg==)
Two halves of a round homogeneous cylinder are held together by a thread wrapped round the cylinder with two equal weights. The complete cylinder weighs 31.4 kg. The plane of contact of both of its halves is vertical. For equilibrium of both haves of the cylinder, the minimum value of m is
![](data:image/png;base64,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)
physics-General
physics-
An unstretched massless rope is wrapped around a solid cylinder of mass
and radius
of the rope is fixed while cylinder is released from rest. The acceleration of the rope is
![](data:image/png;base64,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)
An unstretched massless rope is wrapped around a solid cylinder of mass
and radius
of the rope is fixed while cylinder is released from rest. The acceleration of the rope is
![](data:image/png;base64,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)
physics-General
physics-
A hollow sphere is released from the top of a wedge as shown in the figure. There is no friction between the wedge and the ground. There is sufficient friction to provide pure rolling of the sphere. The velocity of the sphere with respect to ground just before it leaves the wedge will be [Assume masses of the wedge and sphere are equal, h>>R the radius of sphere and the portion of wedge where the sphere leaves the wedge is horizontal]
![](data:image/png;base64,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)
A hollow sphere is released from the top of a wedge as shown in the figure. There is no friction between the wedge and the ground. There is sufficient friction to provide pure rolling of the sphere. The velocity of the sphere with respect to ground just before it leaves the wedge will be [Assume masses of the wedge and sphere are equal, h>>R the radius of sphere and the portion of wedge where the sphere leaves the wedge is horizontal]
![](data:image/png;base64,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)
physics-General
Physics-
A non homogenous solid cylinder of radius r is placed on a rough inclined plane of angle of inclination 60° The angle of repose corresponds to 30°. The maximum radius of gyration in order to prevent slipping is :
A non homogenous solid cylinder of radius r is placed on a rough inclined plane of angle of inclination 60° The angle of repose corresponds to 30°. The maximum radius of gyration in order to prevent slipping is :
Physics-General
physics-
Four solid spheres P, Q, R and S are made to move on a horizontal surface which is rough enough. Sphere P is given a spin and released sphere Q is given a linear velocity and released ; sphere R is given a reverse spin and a forward linear velocity and released ; sphere S is given a forward spin and a forward linear velocity as shown in figure the direction of frictional force F for P, Q, R and S will be along
![](data:image/png;base64,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)
Four solid spheres P, Q, R and S are made to move on a horizontal surface which is rough enough. Sphere P is given a spin and released sphere Q is given a linear velocity and released ; sphere R is given a reverse spin and a forward linear velocity and released ; sphere S is given a forward spin and a forward linear velocity as shown in figure the direction of frictional force F for P, Q, R and S will be along
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)
physics-General
physics-
A disc of radius 20 cm is rolling with slipping on a flat horizontal surface. At a certain instant the velocity of its centre is 4 m/s and its angular velocity is 10rad/s. The lowest contact point is O
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)
A disc of radius 20 cm is rolling with slipping on a flat horizontal surface. At a certain instant the velocity of its centre is 4 m/s and its angular velocity is 10rad/s. The lowest contact point is O
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)
physics-General