","comment":{"@type":"Comment","text":"In order to find the coercion or force from the left end, use equation $F=ma$ and $\\tau =I\\alpha $. The variables of circular motion equation i.e. a and α, can also be used to find the force by equating the two variables."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" $$mg\\dfrac{l}{2}=\\left( \\dfrac{m{{l}^{2}}}{3} \\right)\\alpha $$","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" $$\\alpha =\\dfrac{3g}{2l}$$","position":1},{"@type":"Answer","encodingFormat":"text/html","text":" $$mg-F=ma$$","position":3}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" $$2a=\\dfrac{3g}{2}$$","position":2,"answerExplanation":{"@type":"Comment","text":" $$\\text{Let the desired force from the left support be F, and the downward acceleration of the stick's CM be a}\\text{.} \\\\ \\text{Using equations},\\text{ }F=ma\\text{ and }\\tau =I\\alpha , \\\\ \\text{along with the circular motion relation between }a\\text{ and }\\alpha ,\\text{ we get,} \\\\ mg-F=ma\\text{ }...\\left( 1 \\right) \\\\ mg\\dfrac{l}{2}=\\left( \\dfrac{m{l^2}}{3} \\right)\\alpha \\text{ }...\\left( 2 \\right)\\text{ (using relation of }mgd\\text{ in terms of circular motion)} \\\\ a=\\dfrac{1}{2}l \\alpha \\text{ }...\\left( 3 \\right)\\text{ (}\\because a=r\\alpha \\text{ and }r=\\dfrac{1}{2}d) \\\\ \\text{From equation }\\left( 2 \\right)\\text{ we get}, \\\\ \\Rightarrow \\alpha =\\dfrac{1}{2}mgl\\dfrac{3}{m{l^2}}=\\dfrac{3g}{2l} \\\\ \\text{Equation }\\left( 3 \\right)\\text{ gives}, \\\\ \\Rightarrow a =\\dfrac{3g}{4} \\\\ \\text{and equation }\\left( 1 \\right)\\text{ then gives}, \\\\ \\Rightarrow F=\\dfrac{mg}{4} \\\\ \\text{Note that the right end of the stick accelerates at,} \\\\ 2a=\\dfrac{3g}{2},\\text{which is larger than }g. \\\\ $$","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Angular momentum Quiz 1","text":" A tubular slab of mass $$m$$, radius $$r$$, and moment of Inertia $$I=\\dfrac{1}{2}m{{r}^{2}}$$ pivots without slipping down a plane inclined at an angle $$\\theta $$. What is the gain in propulsion of the center of the slab?
","comment":{"@type":"Comment","text":"We’ll use conservation of energy to determine the speed $v$ of the center of the cylinder after it has moved a distance $d$ down the plane, and then we’ll read off $a$ from the standard constant-acceleration kinematic relation $v=\\sqrt{2ad}$. "},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" $$\\omega =\\dfrac{v}{r}$$","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" $$\\dfrac{m{{v}^{2}}}{2}+\\dfrac{I{{\\omega }^{2}}}{2}$$","position":1},{"@type":"Answer","encodingFormat":"text/html","text":" $$v=\\omega r$$","position":2}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" $$v=\\sqrt{\\left( \\dfrac{4}{3} \\right)gd.\\sin \\theta }$$","position":3,"answerExplanation":{"@type":"Comment","text":" $$\\text{The loss in potential energy of the slab is,} \\\\ \\Delta U=mgd\\sin \\theta \\\\ \\text{Here, }d\\text{ refers to the distance between the slab and the mass}\\text{.} \\\\ \\text{It shows up as kinetic energy},\\text{which equals,} \\\\ K.E=\\dfrac{m{{v}^{2}}}{2}+\\dfrac{I{{\\omega }^{2}}}{2} \\\\ \\text{But the non-slipping condition is,} \\\\ v=\\omega r \\\\ \\Rightarrow \\omega =\\dfrac{v}{r} \\\\ \\text{And},\\text{conservation of energy gives,} \\\\ mgd\\sin \\theta =\\dfrac{1}{2}m{{v}^{2}}+\\dfrac{1}{2}I{{\\omega }^{2}} \\\\ \\Rightarrow mgd\\sin \\theta =\\dfrac{1}{2}m{{v}^{2}}+\\dfrac{1}{2}\\left( \\dfrac{1}{2}m{{r}^{2}} \\right){{\\left( \\dfrac{v}{r} \\right)}^{2}} \\\\ \\Rightarrow mgd\\sin \\theta =\\dfrac{3}{4}m{{v}^{2}} \\\\ \\text{So},\\text{the speed as a function of distance is}, \\\\ v=\\sqrt{\\left( \\dfrac{4}{3} \\right)gd.\\sin \\theta } \\\\ \\text{Hence},\\text{ }v=\\sqrt{2ad} \\\\ \\text{It gives }a=\\left( \\dfrac{2}{3} \\right)g\\sin \\theta \\text{ which is independent of }r. \\\\ $$
","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Angular momentum Quiz 1","text":" A hammer having mass $$m$$ and length $$l$$ is made to hit a rod. The blow is made perpendicular to the rod at one end. Assume the blow occurs quickly, so that the rod doesn’t have time to move much while the hammer is in contact. If the $$CM$$ of the rod ends up moving at speed $$v$$, what will be the pace of the ends right after the blow? ","comment":{"@type":"Comment","text":" In the question, the pace refers to the velocities at the end of the rod. Remember that the angular velocity and the linear velocity are analogous to each other and the relation between them can be found by using the following equation: $\\left( \\dfrac{m{{l}^{2}}}{12} \\right)\\omega =\\left( \\dfrac{l}{2} \\right)mv$. Also, consider the translational circular motion to find the velocities at the ends after the blow, by adding up the values of the rotational motion. "},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" $$\\omega =\\dfrac{6v}{l}$$","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" $$\\left( \\dfrac{m{{l}^{2}}}{12} \\right)\\omega =\\left( \\dfrac{l}{2} \\right)mv$$","position":1},{"@type":"Answer","encodingFormat":"text/html","text":" $$\\Delta L=\\left( \\dfrac{l}{2} \\right)\\Delta p$$","position":2}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" $$v+3v=4v$$","position":3,"answerExplanation":{"@type":"Comment","text":" $$\\text{The equation where we have chosen our origin to be CM is given by,} \\\\ \\Delta L=\\left( \\dfrac{l}{2} \\right)\\Delta \\text{p }\\!\\!~\\!\\!\\text{ } \\\\ \\text{which gives a lever arm of }\\dfrac{l}{2} \\\\ \\text{Therefore, }\\left( \\dfrac{m{{l}^{2}}}{12} \\right)\\omega =\\left( \\dfrac{l}{2} \\right)mv \\\\ \\text{So, the final }v\\text{ and }\\omega \\text{ are related by},\\text{ }\\omega =\\dfrac{6v}{l} \\\\ \\text{The velocities of the ends right after the blow are obtained by adding }\\left( \\text{or subtracting} \\right)\\text{ the rotational motion to } \\\\ \\text{the CMs translational motion}\\text{.} \\\\ \\text{The rotational velocities of the ends relative to the CM are,} \\\\ \\pm \\omega \\left( \\dfrac{l}{2} \\right)\\text{=}\\pm \\left( \\dfrac{6v}{l} \\right)\\left( \\dfrac{l}{2} \\right)\\text{=}\\pm \\text{3}v\\text{ }\\!\\!~\\!\\!\\text{ } \\\\ \\text{Therefore, the end that is hit, moves with velocity }v+3v=4v\\text{, and the other end moves with } \\\\ \\text{velocity }v-3v=-2v\\left( \\text{that is, backward} \\right)\\text{.}$$","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]}]}