Question

A particle is moving along a circular path of radius $5\,m$ with a uniform speed of $5\,m{{s}^{-1}}$. What will be the average acceleration when the particle completes half revolution ?

Answer 1

Acceleration is the rate of change of velocity or the amount of velocity changed in a given amount of time. Average acceleration is the average of change of velocity or the average of different accelerations.

Formula used:
The formula for average acceleration is,
$a=\dfrac{v-u}{t}$
where $v$ is final velocity, $u$ is the initial velocity and $t$ is the time taken.

Complete step by step answer:
The distance covered during the half revolution would be equal to half the perimeter of the full circle. Therefore, the distance covered during half revolution is $\pi r$ where $r=5m$.
And speed = $5\,m{{s}^{-1}}$
Now to find the acceleration we need to find the velocity. If the particle is moving with a constant speed, its speed at initial and final points would be the same.

But the velocity at the final position would be negative of that at the initial position. If the particle starts to move upward at the initial position it would be going downwards at the final position.Hence the direction at final position would be opposite of the position at initial point. Therefore, the initial velocity is $5m{{s}^{-1}}$ and the final velocity is $-5m{{s}^{-1}}$.

And time taken is given by $\dfrac{\text{distance}}{\text{speed}}=\text{time}$
Putting the values in the formula $\frac{\pi \times 5}{5}$
$\text{t= }\pi \text{s}$
Putting the values in $a=\dfrac{v-u}{t}$
$a=\dfrac{5-(-5)m{{s}^{-1}}}{\pi \text{ s}}$
$\therefore a=\dfrac{10}{\pi }\text{ m}{{\text{s}}^{-2}}$

Thus, the average acceleration when the particle completes half revolution is $\dfrac{10}{\pi }\,m{{s}^{-2}}$.

Note: Distance is the actual path covered by the object whereas the displacement is the shortest path between the initial and the final position. Speed is the magnitude of the rate at which the object moved along that path whereas the velocity tells us about the magnitude as well as the direction of the object. Acceleration is the rate of change of velocity.