Rotational Kinetic Energy Practice

Sample questions

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1. Sample 1difficulty 3/5

   A solid sphere rolls without slipping. The fraction of its KE that is <strong>translational</strong> is

   • A

     $2/7$

   • B

     $1/2$

   • C

     $5/7$

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   • D

     $2/5$

   Why

   Trans: $\tfrac{1}{2}M v^2$, Rot: $\tfrac{1}{5}M v^2$, Total $\tfrac{7}{10}M v^2$. Trans fraction: $5/7$.

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2. Sample 2difficulty 3/5

   A solid sphere ($I = \tfrac{2}{5}M R^2$) rolls without slipping at speed $v$. The fraction of its total KE that is <strong>rotational</strong> is

   • A

     $2/5$

   • B

     $1/2$

   • C

     $2/7$

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   • D

     $1/5$

   Why

   Trans: $\tfrac{1}{2}M v^2$. Rot: $\tfrac{1}{2}\cdot\tfrac{2}{5}M R^2 (v/R)^2 = \tfrac{1}{5}M v^2$. Total $\tfrac{7}{10}M v^2$. Rotational fraction: $(1/5)/(7/10) = 2/7$.

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3. Sample 3difficulty 3/5

   A bowling ball ($I = \tfrac{2}{5}MR^2$) rolls at $5~\text{m/s}$ along a frictionless lane (no slipping). Its mass is $7~\text{kg}$. Its <strong>total</strong> kinetic energy is

   • A

     $122.5~\text{J}$

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   • B

     $245~\text{J}$

   • C

     $87.5~\text{J}$

   • D

     $175~\text{J}$

   Why

   $K = \tfrac{7}{10}M v^2 = 0.7(7)(25) = 122.5~\text{J}$.

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4. Sample 4difficulty 4/5

   A solid sphere rolls without slipping at speed $v$. What fraction of its total kinetic energy is rotational?

   • A

     $2/7$

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   • B

     $1/2$

   • C

     $5/7$

   • D

     $2/5$

   Why

   $KE_{\text{trans}} = \tfrac{1}{2}m v^2$ and $KE_{\text{rot}} = \tfrac{1}{2}I\omega^2 = \tfrac{1}{2}(\tfrac{2}{5}m R^2)(v/R)^2 = \tfrac{1}{5}m v^2$. Total $= \tfrac{7}{10}mv^2$. Rotational fraction $= (1/5)/(7/10) = 2/7$.

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5. Sample 5difficulty 4/5

   A thin hoop ($I = MR^2$) rolls without slipping on a level surface with center-of-mass speed $v$. What fraction of its total kinetic energy is rotational?

   • A

     $2/5$

   • B

     $2/7$

   • C

     $1/2$

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   • D

     $1/3$

   Why

   $K_{trans} = \tfrac{1}{2}Mv^2$ and $K_{rot} = \tfrac{1}{2}(MR^2)(v/R)^2 = \tfrac{1}{2}Mv^2$. So $K_{rot}/K_{tot} = 1/2$.

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