Chapter 9, Problem 6STP

To determine

To choose:The correct option.

Answer to Problem 6STP

The correct option is (C), $-8L_{\text{Small}}$ .

Explanation of Solution

Given:

The moment of inertia for a disk is $\left(\frac{1}{2}\right)m{r}^2$ .

The mass of the larger gear is $m_{\text{Large}}=4m_{\text{Small}}$ .

The radius of the larger gear is $r_{\text{Large}}=2r_{\text{Small}}$ .

Formula used:

The expression for the angular momentum is,

  $L=I\omega$

Here, $I$ is the moment of inertia and $\omega$ is the angular velocity.

The expression for the angular velocity is,

  $\omega =\frac{v}{r}$

Here, $v$ is the linear velocity and $r$ is the radius.

Calculation:

The linear speed of the large gear using the relation of the angular momentum in the large gear is,

$\begin{array}{l}{L}_{\text{Large}}=I_{\text{Large}}{\omega }_{\text{Large}}\\ L_{\text{Large}}=-\left(\frac{1}{2}{m}_{\text{Large}}{r}_{\text{Large}}^2\right)\left(\frac{v_{\text{Large}}}{r_{\text{Large}}}\right)\\ L_{\text{Large}}=-\left(\frac{1}{2}{m}_{\text{Large}}{r}_{\text{Large}}\right)\left(v_{\text{Large}}\right)\\ v_{\text{Large}}=-\frac{2L_{\text{Large}}}{m_{\text{Large}}{r}_{\text{Large}}}\end{array}$

The linear speed of the small gear using the relation of the angular momentum in the small gear is,

$\begin{array}{l}{L}_{\text{Small}}=I_{\text{Small}}{\omega }_{\text{Small}}\\ L_{\text{Small}}=\left(\frac{1}{2}{m}_{\text{Small}}{r}_{\text{Small}}^2\right)\left(\frac{v_{\text{Small}}}{r_{\text{Small}}}\right)\\ L_{\text{Small}}=\left(\frac{1}{2}{m}_{\text{Small}}{r}_{\text{Small}}\right)\left(v_{\text{Small}}\right)\\ v_{\text{Small}}=\frac{2L_{\text{Small}}}{m_{\text{Small}}{r}_{\text{Small}}}\end{array}$

The linear speed of the large gear is equal to the linear speed of the small gear.

$\begin{array}{l}{v}_{\text{Large}}=v_{\text{Small}}\\ -\frac{2L_{\text{Large}}}{m_{\text{Large}}{r}_{\text{Large}}}=\frac{2L_{\text{Small}}}{m_{\text{Small}}{r}_{\text{Small}}}\\ -\frac{L_{\text{Large}}}{4m_{\text{Small}}\times 2r_{\text{Small}}}=\frac{L_{\text{Small}}}{m_{\text{Small}}{r}_{\text{Small}}}\\ L_{\text{Large}}=-8L_{\text{Small}}\end{array}$

Conclusion:

Thus, the angular momentum of the larger gear as a function of the angular momentum of the smaller gear is option (C), $-8L_{\text{Small}}$ .