a student on a piano stool rotates freely with an angular speed of $2.85\text{ rev/s}$. the student holds a $1.45\text{ kg}$ mass in each outstretched arm, $0.749\text{ m}$ from the axis of rotation. the combined moment of inertia of the student and the stool, ignoring the two masses, is $5.53\text{ kg}\cdot\text{m}^2$, a value that remains constant.

part a
as the student pulls his arms inward, his angular speed increases to $3.44\text{ rev/s}$. how far are the masses from the axis of rotation at this time, considering the masses to be points?
$d = $ m

part b
calculate the initial kinetic energy of the system.
$k_i = $ j

part c
calculate the final kinetic energy of the system.

Question

a student on a piano stool rotates freely with an angular speed of $2.85\text{ rev/s}$. the student holds a $1.45\text{ kg}$ mass in each outstretched arm, $0.749\text{ m}$ from the axis of rotation. the combined moment of inertia of the student and the stool, ignoring the two masses, is $5.53\text{ kg}\cdot\text{m}^2$, a value that remains constant.

part a
as the student pulls his arms inward, his angular speed increases to $3.44\text{ rev/s}$. how far are the masses from the axis of rotation at this time, considering the masses to be points?
$d = $ m

part b
calculate the initial kinetic energy of the system.
$k_i = $ j

part c
calculate the final kinetic energy of the system.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Conservation of Angular Momentum", "Moment of Inertia" ], "new_concepts": [], "current_concepts": [ "Conservation of Angular Momentum", "Moment of Inertia", "Rotational Kinetic Energy" ] } ## Calculate the initial moment of inertia and initial angular velocity $$ \begin{aligned} &I_{\text{stool}} = 5.53\text{ kg}\cdot\text{m}^2\ &m = 1.45\text{ kg},\quad r_i = 0.749\text{ m}\ &I_i = I_{\text{stool}} + 2 \cdot m \cdot r_i^2 = 5.53 + 2 \cdot 1.45 \cdot 0.749^2 = 5.53 + 1.6269 = 7.1569\text{ kg}\cdot\text{m}^2\ &\omega_i = 2.85\text{ rev/s} = 2.85 \cdot 2\pi\text{ rad/s} \approx 17.907\text{ rad/s} \end{aligned} $$ ## Solve for the final distance of the masses from the axis of rotation (Part A) $$ \begin{aligned} &\omega_f = 3.44\text{ rev/s}\ &I_i \omega_i = I_f \omega_f \implies I_f = I_i \frac{\omega_i}{\omega_f} = 7.1569 \cdot \frac{2.85}{3.44} \approx 5.9311\text{ kg}\cdot\text{m}^2\ &I_f = I_{\text{stool}} + 2 \cdot m \cdot d^2 \implies 5.9311 = 5.53 + 2 \cdot 1.45 \cdot d^2\ &2.90 \cdot d^2 = 0.4011 \implies d^2 \approx 0.1383 \implies d \approx 0.372\text{ m} \end{aligned} $$ ## Calculate the initial and final kinetic energies (Part B and Part C) $$ \begin{aligned} &K_i = \frac{1}{2} I_i \omega_i^2 = \frac{1}{2} \cdot 7.1569 \cdot (17.907)^2 \approx 1147.5\text{ J} \approx 1.15 \times 10^3\text{ J}\ &\omega_f = 3.44\text{ rev/s} = 3.44 \cdot 2\pi\text{ rad/s} \approx 21.614\text{ rad/s}\ &K_f = \frac{1}{2} I_f \omega_f^2 = \frac{1}{2} \cdot 5.9311 \cdot (21.614)^2 \approx 1385.4\text{ J} \approx 1.39 \times 10^3\text{ J} \end{aligned} $$ ## Question 1 $d = 0.372\text{ m}$ ## Question 2 $K_i = 1.15 \times 10^3\text{ J}$ ## Question 3 $K_f = 1.39 \times 10^3\text{ J}$ { "subject": "Natural Science", "question_type": "Multi-part", "knowledge_point": [ "Natural Science", "Physics", "Conservation of Angular Momentum" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the initial moment of inertia and initial angular velocity

Solve for the final distance of the masses from the axis of rotation (Part A)

Calculate the initial and final kinetic energies (Part B and Part C)

Answer:

Question 1

Question 2

Question 3