ip a turntable with a moment of inertia of $5.0 \times 10^{-3} \text{ kg} \cdot \text{m}^2$ rotates freely with an angular speed of $33\frac{1}{3} \text{ rpm}$. riding on the rim of the turntable, $20 \text{ cm}$ from the center, is a cute, $30 \text{ g}$ mouse.

part a
if the mouse walks to the center of the turntable, will the turntable rotate faster, slower, or at the same rate?
- faster
- slower
- at the same rate

part c
calculate the angular speed of the turntable when the mouse reaches the center.
express your answer using two significant figures.
$\omega_f =$ $\text{rad/s}$

Question

ip a turntable with a moment of inertia of $5.0 \times 10^{-3} \text{ kg} \cdot \text{m}^2$ rotates freely with an angular speed of $33\frac{1}{3} \text{ rpm}$. riding on the rim of the turntable, $20 \text{ cm}$ from the center, is a cute, $30 \text{ g}$ mouse.

part a
if the mouse walks to the center of the turntable, will the turntable rotate faster, slower, or at the same rate?
- faster
- slower
- at the same rate

part c
calculate the angular speed of the turntable when the mouse reaches the center.
express your answer using two significant figures.
$\omega_f =$   $\text{rad/s}$

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Conservation of Angular Momentum", "Moment of Inertia", "Unit Conversion" ], "new_concepts": [], "current_concepts": [ "Conservation of Angular Momentum", "Moment of Inertia", "Rotational Kinematics", "Unit Conversion" ] } ## Identify given values and convert units $$ \begin{aligned} I_{\text{disk}} &= 5.0 \times 10^{-3} \text{ kg}\cdot\text{m}^2 \ \omega_i &= 33\frac{1}{3} \text{ rpm} = \frac{100}{3} \times \frac{2\pi}{60} \text{ rad/s} = \frac{10\pi}{9} \text{ rad/s} \approx 3.4907 \text{ rad/s} \ m &= 30 \text{ g} = 0.030 \text{ kg} \ r &= 20 \text{ cm} = 0.20 \text{ m} \end{aligned} $$ ## Calculate initial and final moments of inertia $$ \begin{aligned} I_i &= I_{\text{disk}} + m r^2 = 5.0 \times 10^{-3} + 0.030 \times (0.20)^2 = 5.0 \times 10^{-3} + 1.2 \times 10^{-3} = 6.2 \times 10^{-3} \text{ kg}\cdot\text{m}^2 \ I_f &= I_{\text{disk}} = 5.0 \times 10^{-3} \text{ kg}\cdot\text{m}^2 \end{aligned} $$ ## Apply conservation of angular momentum to solve for final angular speed $$ \begin{aligned} I_i \omega_i &= I_f \omega_f \ \omega_f &= \frac{I_i}{I_f} \omega_i = \frac{6.2 \times 10^{-3}}{5.0 \times 10^{-3}} \times \frac{10\pi}{9} = 1.24 \times 3.4907 \approx 4.328 \text{ rad/s} \approx 4.3 \text{ rad/s} \end{aligned} $$ ## Question 1 faster

slower

at the same rate ## Question 2 $\omega_f =$ $4.3$ $\text{rad/s}$ { "subject": "Natural Science", "question_type": "Multi-part", "knowledge_point": [ "Natural Science", "Physics", "Conservation of Angular Momentum" ] }

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QUESTION IMAGE

Question

Explanation:

Identify given values and convert units

Calculate initial and final moments of inertia

Apply conservation of angular momentum to solve for final angular speed

Answer:

Question 1

Question 2