a child exerts a tangential (53.4\text{ n}) force on the rim of a disk-shaped merry-go-round with a radius of (2.79\text{ m}).

for the steps and strategies involved in solving a similar problem, you may view the following quick example 11-22 video:

part a

if the merry-go-round starts at rest and acquires an angular speed of (0.9250\text{ rev/s}) in (3.00\text{ s}), what is its mass?

express your answer in kilograms.

Question

a child exerts a tangential (53.4\text{ n}) force on the rim of a disk-shaped merry-go-round with a radius of (2.79\text{ m}).

for the steps and strategies involved in solving a similar problem, you may view the following quick example 11-22 video:

part a

if the merry-go-round starts at rest and acquires an angular speed of (0.9250\text{ rev/s}) in (3.00\text{ s}), what is its mass?

express your answer in kilograms.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Torque", "Rotational Kinematics", "Rotational Dynamics", "Moment of Inertia" ], "new_concepts": [ "Rotational Inertia of a Disk" ], "current_concepts": [ "Torque", "Rotational Kinematics", "Rotational Dynamics", "Moment of Inertia" ] } ## Calculate the net torque on the merry-go-round Using the *Torque* knowledge point $$ \tau = F \cdot R = 53.4\text{ N} \times 2.79\text{ m} = 148.986\text{ N}\cdot\text{m} $$ ## Determine the angular acceleration Using the *Rotational Kinematics* knowledge point $$ \begin{aligned} \omega_0 &= 0\text{ rad/s}\ \omega &= 0.9250\text{ rev/s} \times 2\pi\text{ rad/rev} = 1.85\pi\text{ rad/s} \approx 5.81195\text{ rad/s}\ \alpha &= \frac{\omega - \omega_0}{t} = \frac{1.85\pi\text{ rad/s}}{3.00\text{ s}} \approx 1.93732\text{ rad/s}^2 \end{aligned} $$ ## Relate torque to moment of inertia Using the *Rotational Dynamics* knowledge point $$ \begin{aligned} \tau &= I \cdot \alpha\ I &= \frac{\tau}{\alpha} = \frac{148.986\text{ N}\cdot\text{m}}{1.93732\text{ rad/s}^2} \approx 76.9033\text{ kg}\cdot\text{m}^2 \end{aligned} $$ ## Calculate the mass of the disk Using the *Moment of Inertia* knowledge point $$ \begin{aligned} I &= \frac{1}{2} M R^2\ M &= \frac{2I}{R^2} = \frac{2 \times 76.9033\text{ kg}\cdot\text{m}^2}{(2.79\text{ m})^2} \approx 19.759\text{ kg} \end{aligned} $$ Rounding to three significant figures gives $19.8\text{ kg}$. If the merry-go-round starts at rest and acquires an angular speed of 0.9250 rev/s in 3.00 s, what is its mass? Express your answer in kilograms. $m =$ $19.8$ kg { "subject": "Natural Science", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Natural Science", "Physics", "Rotational Dynamics" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Calculate the net torque on the merry-go-round

Determine the angular acceleration

Relate torque to moment of inertia

Calculate the mass of the disk

Answer: