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Rolling

AP Physics 1· difficulty 4/5

θ R, M h

A solid uniform sphere of mass MM and radius RR rolls without slipping from rest down an incline through a vertical drop hh. What is the speed of the center of mass at the bottom?

  • A

    10gh7\sqrt{\tfrac{10gh}{7}}

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  • B

    2gh\sqrt{2gh}

  • C

    6gh5\sqrt{\tfrac{6gh}{5}}

  • D

    4gh3\sqrt{\tfrac{4gh}{3}}

Explanation

Energy conservation: Mgh=12Mv2+12Iω2Mgh = \tfrac{1}{2}Mv^2 + \tfrac{1}{2}I\omega^2 with I=25MR2I = \tfrac{2}{5}MR^2 and ω=v/R\omega = v/R gives Mgh=710Mv2Mgh = \tfrac{7}{10}Mv^2, so v=10gh/7v = \sqrt{10gh/7}.

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