Chapter 8.2, Problem 28PP

To determine

The force needed to exert on the string.

Answer to Problem 28PP

The force needed to exert on the string is $4.3\text{ N}$ .

Explanation of Solution

Given:

The radius of the gear is $r_{\text{gear}}=0.11\text{ m}$ .

The radius of the wheel is $r_{\text{wheel}}=0.38\text{ m}$ .

The force on the chain is $F_{\text{chain}}=15\text{ N}$ .

Formula used:

The expression for the torque is,

$\tau =Fr$

Here, $F$ is the force and $r$ is the radial distance.

Calculation:

Consider the torque on the wheel comes from either the string or the chain. Hence, the torque ${\tau }_{\text{wheel}}={\tau }_{\text{chain}}$ .

The force on the string is,

$\begin{array}{l}{\tau }_{\text{wheel}}={\tau }_{\text{chain}}\\ F_{\text{chain}}{r}_{\text{gear}}=F_{\text{string}}{r}_{\text{wheel}}\\ \left(15\text{ N}\right)\left(0.11\text{ m}\right)=F_{\text{string}}\left(0.38\text{ m}\right)\\ F_{\text{string}}=4.3\text{ N}\end{array}$

Conclusion:

Thus, the force needed to exert on the string is $4.3\text{ N}$ .