Question

A stone tied to the end of a string 80cm long is whirled in a horizontal circle at a constant speed. If the stone makes 14 revolutions in 25s, what is the magnitude and direction of the acceleration of the stone?

Answer 1

Recall that the rate of change of velocity with time is known as acceleration. Since it is a vector quantity, it will have both magnitude and direction. When an object moves in a rotational motion it is said to have rotational acceleration.

Complete step by step answer:
Step I:
Given the length of the string is $80cm = 0.8m$
It makes $14$ revolutions in $25\sec$
Frequency is defined as the number of events that are occurring in a given period of time. Its formula is
$f = \dfrac{{\text{No. of revolutions}}}{{\text{time}}}$
Frequency of the string will be $f = \dfrac{{14}}{{25}}$
Step II:
Angular speed is defined as the rate of change of angular displacement. In one rotation, angular distance is $2\pi$ and the time period is $T$.
Angular speed is given by formula
$\Rightarrow \omega = \dfrac{{2\pi }}{t}$
Since it is known that
$\Rightarrow \dfrac{1}{t} = f$
$\Rightarrow \omega = 2\pi f$
Where $\omega$ is the angular speed
$f$ is the frequency
Step III:
Substituting values in the above formula,
$\Rightarrow \omega = 2\pi \times \dfrac{{14}}{{25}}$
$\Rightarrow \omega = \dfrac{{28\pi }}{{25}}rad/\sec$
Step IV:
Also acceleration of a rotating body is given by
$a = {\omega ^2}r$
$\omega$ is the angular speed of the stone
$r$ is the radius
$\Rightarrow a = 0.8 \times {(\dfrac{{28\pi }}{{25}})^2}$
$\Rightarrow a = 0.8 \times \dfrac{{784 \times 9.859}}{{625}} = \dfrac{{6183.564}}{{625}}$
$\Rightarrow a = 9.89m/{s^2}$

The magnitude of the acceleration of the stone is $9.89m/{s^2}$ and the direction will be along the radius at every point towards the center.

Note:
It is to be noted that sometimes angular velocity can be constant. The direction of the vector is the same as that of the direction of motion of the particle. When an object is moving with constant velocity then the particle is experiencing a centripetal force. The particle is then said to be moving with constant speed and angular rate of rotation. Therefore the particle will have a constant acceleration.