Chapter 8, Problem 61P

(a)

To determine

The angular velocity.

Answer to Problem 61P

The angular velocity of the platform when person reaches the edge is $1.32\text{rad/s}$

Explanation of Solution

Given:

Moment of inertia of merry-go-round, $I_1=920kg-m^2$

Mass of person, $M=75kg$

Radius of merry-go-round, $R=3.0m$

Initial angular velocity of platform, ${\omega }_1=2\text{rad/s}$

Formula used:

Moment of inertia of person, $I=M{R}^2$

Law of conservation of angular momentum, $I_1{\omega }_1=(I_1+I_2)\omega \text{'}$

Calculation:

Substituting values for moment of inertia of person,

  $\begin{array}{l}$ I_2=(75){(3)}^2$I_2=675kg-m^2\end{array}$

Substituting values in law of conservation of angular momentum,

  $\begin{array}{l}(920)(2)=(920+675)\omega \text{'}\\ \omega \text{'}=\frac{1840}{1395}\\ \omega \text{'}=1.32\text{rad/s}\end{array}$

Conclusion:

The angular velocity of the platform when person reaches the edge is $1.32\text{rad/s}$

(b)

To determine

The rotational kinetic energy of the system.

Answer to Problem 61P

The rotational kinetic energy of the platform before the person walks is $1840J$

The rotational kinetic energy of the platform after the person walks is $1390J$

Explanation of Solution

Given:

Moment of inertia of merry-go-round, $I_1=920kg-m^2$

Moment of inertia of person, $I_2=675kg-m^2$

Initial angular velocity of platform, ${\omega }_1=2\text{rad/s}$

Final angular velocity of platform, $\omega \text{'}=1.32\text{rad/s}$

Formula used:

Rotational kinetic energy is given by, $K=\frac{1}{2}I{\omega }^2$

Calculation:

Substituting values for rotational kinetic energy before man walks,

  $\begin{array}{l}{K}_1=\frac{1}{2}{I}_1{\omega }_1^2\\ $ K_1=\frac{1}{2}(920){(2)}^2$K_1=1840J\end{array}$

Substituting values for rotational kinetic energy after man walks,

  $\begin{array}{l}$ K_2=\frac{1}{2}(I_1+I_2){(\omega \text{'})}^2$$ K_2=\frac{1}{2}(920+675){(1.32)}^2$K_2=1390J\end{array}$

Conclusion:

The rotational kinetic energy of the platform before the person walks is $1840J$

The rotational kinetic energy of the platform after the person walks is $1390J$