question 8

a different scientist has some device of mysterious purpose. the device has a large horizontal disk ($m = 100\text{ kg}$, $i = 1/2\,mr^2$). on the very edge of the disk sits a large child ($m = 40\text{ kg}$, $i = mr^2$) riding on the disk. the disk is spinning at $0.5\text{ radians per second}$.

after a while, the child jumps off the disk, and the disk speeds up. what is the disks final angular velocity?

Question

question 8

a different scientist has some device of mysterious purpose. the device has a large horizontal disk ($m = 100\text{ kg}$, $i = 1/2\,mr^2$). on the very edge of the disk sits a large child ($m = 40\text{ kg}$, $i = mr^2$) riding on the disk. the disk is spinning at $0.5\text{ radians per second}$.

after a while, the child jumps off the disk, and the disk speeds up. what is the disks final angular velocity?

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Conservation of Angular Momentum", "Rotational Inertia of a Disk" ], "new_concepts": [ "Angular Momentum Conservation with Mass Loss" ], "current_concepts": [ "Conservation of Angular Momentum", "Rotational Inertia of a Disk", "Angular Momentum Conservation with Mass Loss" ] } ## Calculate initial moment of inertia $$ \begin{aligned} I_{\text{disk}} &= \frac{1}{2} M R^2 = \frac{1}{2} (100) R^2 = 50 R^2 \ I_{\text{child}} &= m R^2 = 40 R^2 \ I_i &= I_{\text{disk}} + I_{\text{child}} = 50 R^2 + 40 R^2 = 90 R^2 \end{aligned} $$ ## Apply conservation of angular momentum $$ \begin{aligned} L_i &= L_f \ I_i \omega_i &= I_f \omega_f \ (90 R^2) (0.5) &= (50 R^2) \omega_f \end{aligned} $$ ## Solve for final angular velocity $$ \begin{aligned} 45 R^2 &= 50 R^2 \omega_f \ \omega_f &= \frac{45}{50} = 0.9\text{ rad/s} \end{aligned} $$ After a while, the child jumps off the disk, and the disk speeds up. What is the disk's final angular velocity? $0.9$ radians per second. { "subject": "Natural Science", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Natural Science", "Physics", "Conservation of Angular Momentum" ] }

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Question

Explanation:

Calculate initial moment of inertia

Apply conservation of angular momentum

Solve for final angular velocity

Answer: