Chapter 10, Problem 28A

A large wheel is coupled to a wheel with half the diameter as shown.

a. How does the rotational speed of the smaller wheel compare with that of the larger wheel?

b. How do the tangential speeds at the rims compare (assuming the belt doesn’t slip)?

To determine

The comparison of the rotational speed of the smaller and the larger wheel shown in the given figure.

Answer to Problem 28A

Same

Explanation of Solution

Introduction:

A linear speed $\left(v\right)$ is written in terms of the rotational speed $\left(\omega \right)$ as,

  $v=r\omega$

Where, $r=$ distance from the axis

Now since both the wheels are coupled using a belt, hence they will rotate with the same speed.

Conclusion:

Hence, the rotational speed of both the wheel will be same.

To determine

The comparison of the tangential speed at the rims of the smaller and the larger wheel shown in the given figure.

Answer to Problem 28A

Tangential speed at the rim will be more for the larger wheel as compared to the smaller wheel.

Explanation of Solution

Introduction:

A linear speed $\left(v\right)$ is written in terms of the rotational speed $\left(\omega \right)$ as,

  $v=r\omega$

Where, $r=$ distance from the axis

Since the rotational speed of both the wheels are same, thus

  $v\propto r$

And, distance of the rims from the axis of rotation is larger for the larger wheel and smaller for the smaller wheel,

  $r_L>r_S$

Thus,

  $v_L>v_S$

Conclusion:

Hence, the larger wheel has more tangential speed at the rims as compared to the smaller wheel.