Chapter 10, Problem 1OQ

To determine

Total distance travelled by the wheel.

Answer to Problem 1OQ

Option(c). Distance travelled by the wheel is $1.00\text{km}$.

Explanation of Solution

The wheel has clicked $320$ times i.e. it has made $320$ revolutions.

Write the expression for the distance travelled.

    $s=r\theta$                                                                                                            (I)

Here, $s$ is the distance travelled, $r$ is the radius and $\theta$ is the angular distance travelled.

Write the expression for angular displacement.

    $\theta =2\pi n$                                                                                                       (II)

Here, $n$  is the number of revolutions.

Conclusion:

Substitute $320\text{rev}$ for $n$ in equation (II).

    $\begin{array}{c}\theta =2\pi (320)\\ =640\pi \text{rad}\end{array}$

Substitute $0.500\text{m}$ for $r$ and $640\pi \text{rad}$ for $\theta$ in equation (I).

    $\begin{array}{c}s=\left(0.500\text{m}\right)\left(640\pi \text{rad}\right)\\ =1005.3\text{m}\\ \text{=}\left(1005.3\text{m}\right)\left(\frac{1\text{km}}{1000\text{m}}\right)\\ =1.0\text{km}\end{array}$

Distance travelled by the wheel is $1.00\text{km}$. Thus, Option(c) is correct.

Distance travelled by the wheel is not equal to $0.50\text{km}$ because distance travelled is not equal to $320\pi r$. Thus; Option (a) is incorrect

Distance travelled by the wheel is not equal to $0.80\text{km}$ as it is proportional to radius and for the given radius this can’t be correct. Thus; Option (b) is incorrect

Distance travelled by the wheel is not equal to $1.5\text{km}$ because distance travelled is equal to $r\theta$ and after substitution this can’t be the answer. Thus; Option (d) is incorrect

Distance travelled by the wheel is not equal to $1.8\text{km}$ as this value is much greater than the correct option which is not possible in the case of values given in the question. Thus; Option (e) is incorrect