Conservation of Energy Practice

Sample questions

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1. Sample 1difficulty 2/5

   A pendulum is released from rest with the string horizontal (length $L$). Using $g \approx 10~\text{m/s}^2$, the bob's speed at the lowest point is

   • A

     $\sqrt{gL}$

   • B

     $\sqrt{2gL}$

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   • C

     $gL$

   • D

     $2\sqrt{gL}$

   Why

   Drop = $L$. $\tfrac{1}{2}v^2 = gL \Rightarrow v = \sqrt{2gL}$.

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2. Sample 2difficulty 2/5

   A $4~\text{kg}$ box slides $5~\text{m}$ across a floor with $\mu_k = 0.30$. Using $g \approx 10~\text{m/s}^2$, the thermal energy generated is

   • A

     $6~\text{J}$

   • B

     $120~\text{J}$

   • C

     $12~\text{J}$

   • D

     $60~\text{J}$

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   Why

   $\Delta E_\text{th} = \mu_k m g d = (0.30)(4)(10)(5) = 60~\text{J}$.

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3. Sample 3difficulty 2/5

   A pendulum is released from rest at point A ($60^\circ$ from vertical) and swings to its lowest point B. Comparing energies at A and B,

   • A

     $U_A > U_B$ but $K_A < K_B$

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   • B

     $U_A = U_B$ and $K_A = K_B$

   • C

     $U_B > U_A$

   • D

     $K_A > K_B$

   Why

   Released from rest at A: $K_A = 0$, $U_A$ maximum. At B (lowest): $U_B$ minimum, $K_B$ maximum.

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4. Sample 4difficulty 2/5

   A frictionless cart starts at rest at height $h_0$. Its speed at height $h < h_0$ is

   • A

     $\sqrt{2 g h}$

   • B

     $\sqrt{2 g (h_0 - h)}$

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   • C

     $\sqrt{2 g h_0}$

   • D

     $\sqrt{2 g (h_0 + h)}$

   Why

   Energy: $\tfrac{1}{2}v^2 = g(h_0 - h)$, so $v = \sqrt{2g(h_0 - h)}$.

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5. Sample 5difficulty 2/5

   An object is released from rest at height $h = 5.0~\text{m}$ (ignore air resistance, $g \approx 10~\text{m/s}^2$). How fast is it moving just before hitting the ground?

   • A

     $10~\text{m/s}$

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   • B

     $7.1~\text{m/s}$

   • C

     $5~\text{m/s}$

   • D

     $50~\text{m/s}$

   Why

   $mgh = \tfrac{1}{2}m v^2 \Rightarrow v = \sqrt{2gh} = \sqrt{2(10)(5)} = 10~\text{m/s}$.

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