Chapter 8, Problem 66P

(a)

To determine

The angular velocity of the platform.

Answer to Problem 66P

The angular velocity of the platform is $\frac{-I_W{\omega }_W}{I_P}$

Explanation of Solution

Given:

Moment of inertia of the person with the platform, $I_P$

Moment of inertia of the wheel, $I_W$

Angular velocity of the wheel, ${\omega }_W$

A person moves the axis of the wheel pointing vertically upwards

Formula used:

Law of conservation of angular momentum,

  $L_1=L_2$

Calculation:

Initially, both the platform and the wheel is at rest.

Hence, initial angular momentum, $L_1=0$

Let final angular velocity be ${\omega }_P$ .

Substituting values in law of conservation of angular momentum,

  $\begin{array}{l}0=I_W{\omega }_W+I_P{\omega }_P\\ {\omega }_P=\frac{-I_W{\omega }_W}{I_P}\end{array}$

Conclusion:

The angular velocity of the platform is $\frac{-I_W{\omega }_W}{I_P}$

(b)

To determine

The angular velocity of the platform.

Answer to Problem 66P

The angular velocity of the platform is $\frac{-I_W{\omega }_W}{2I_P}$ .

Explanation of Solution

Given:

Moment of inertia of the person with the platform, $I_P$

Moment of inertia of the wheel, $I_W$

Angular velocity of the wheel, ${\omega }_W$

A person moves the axis of the wheel at 60° to the vertical.

Formula used:

Law of conservation of angular momentum,

  $L_1=L_2$

Calculation:

Initially, both the platform and the wheel is at rest.

Hence, initial angular momentum, $L_1=0$

Let final angular velocity be ${\omega }_P$ .

Substituting values in law of conservation of angular momentum,

  $\begin{array}{l}0=I_W{\omega }_W\cos 60°+I_P{\omega }_P\\ {\omega }_P=\frac{-I_W{\omega }_W}{2I_P}\end{array}$

Conclusion:

The angular velocity of the platform is $\frac{-I_W{\omega }_W}{2I_P}$

(c)

To determine

The angular velocity of the platform.

Answer to Problem 66P

The angular velocity of the platform is $\frac{I_W{\omega }_W}{I_P}$

Explanation of Solution

Given:

Moment of inertia of the person with the platform, $I_P$

Moment of inertia of the wheel, $I_W$

Angular velocity of the wheel, $-{\omega }_W$

A person moves the axis of the wheel pointing vertically downwards.

Formula used:

Law of conservation of angular momentum,

  $L_1=L_2$

Calculation:

Initially, both the platform and the wheel are at rest.

Hence, initial angular momentum, $L_1=0$

Let final angular velocity be ${\omega }_P$ .

Substituting values in law of conservation of angular momentum,

  $\begin{array}{l}0=-I_W{\omega }_W+I_P{\omega }_P\\ {\omega }_P=\frac{I_W{\omega }_W}{I_P}\end{array}$

Conclusion:

The angular velocity of the platform is $\frac{I_W{\omega }_W}{I_P}$

(d)

To determine

The angular velocity of the platform.

Answer to Problem 66P

The angular velocity of the platform is 0.

Explanation of Solution

Given:

Moment of inertia of the person with the platform, $I_P$

Moment of inertia of the wheel, $I_W$

A person stops rotation of the wheel

Formula used:

Law of conservation of angular momentum,

  $L_1=L_2$

Calculation:

Initially, both the platform and the wheel are at rest.

Hence, initial angular momentum, $L_1=0$

Let final angular velocity be ${\omega }_P$ .

Substituting values in law of conservation of angular momentum,

  $\begin{array}{l}0=I_W(0)+I_P{\omega }_P\\ {\omega }_P=0\end{array}$

Conclusion:

The angular velocity of the platform is $0$