Question

The linear speed of the second’s hand of a wall clock is $1.05\;cm/s$ What will be the length of the second hand?

Answer 1

We know that linear speed is the measure at which the object travels or covers a certain linear distance. And angular speed is the measure at which a rotating object covers a certain circular path. Using the two and their relationship, we can solve the following question as shown below.

Complete step by step solution:
A body which undergoes rotation like the seconds hand here has two kinds of velocity namely the angular velocity$\omega$ and the linear velocity$v$ .The linear velocity is mathematically defined as $v=\dfrac{s}{t}$ where $s$ is the linear distance covered in time $t$ whereas angular velocity is mathematically defined as $\omega=\dfrac{\theta}{t}$ where $\theta$ is the angular distance covered at time $t$ . Also the relationship between the two is given as $v=\omega r$, where $r$ is the radius of the circular path
Since $s=r\theta$ or the length of the arc, $s$ is the product of the radius $r$ and the angle $\theta$ subtended by it in the circle.
Hence solving the above equations, we get$v=\omega r$
Given that the linear speed of the seconds hand is $v=1.05\;cm/s$, then then angular velocity of the seconds hand will be $\omega=\dfrac{2\pi}{60}$,let the length of the second’s hand be $r$, then
Then from $v=\omega r$, substituting the values we have
$\dfrac{2\pi \times r}{60}=1.05$
$\implies r=\dfrac{1.05\times 60}{2\pi}$
$\implies r=\dfrac{63}{2\times 3.14}$
$\therefore r=10.03\;cm$
Thus the length of the seconds hand was found to be $10.03\;cm$

Note: Linear velocity helps in the movement of the object in the forward direction, whereas the angular velocity is due to the centripetal force acting on the rotating object and helps in the tangential direction; hence both are required for the rotating object to be stable.