A particle moves with a constant speed v along a circular path of radius r and completes the circle in time T. What is the acceleration of the particle

The correct option is 2πv/T.

Let us understand the motion of the particle step-by-step.

1. Understanding the type of motion:
The particle moves along a circular path of radius $r$ with a constant speed $v$. When a particle moves in a circle with constant speed, it undergoes uniform circular motion. In this type of motion, even though the speed is constant, the direction of velocity changes continuously. This change in direction gives rise to an acceleration directed towards the center of the circle, known as the centripetal acceleration.

2. Formula for centripetal acceleration:
The magnitude of the centripetal acceleration $a$ is given by the standard formula:
$a=\frac{v^2}{r}$

3. Relating speed, radius, and time period:
The particle completes one full revolution of the circle in time $T$. The distance traveled in one complete revolution is the circumference of the circular path, which is $2\pi r$.
Since speed $v$ is constant, we can express the speed as:
$v=\frac{\text{Distance}}{\text{Time}}=\frac{2\pi r}{T}$

4. Solving for radius $r$:
From the equation for speed, we can express the radius $r$ in terms of speed $v$ and time period $T$:
$r=\frac{vT}{2\pi }$

5. Substituting $r$ back into the acceleration formula:
Now, we substitute this expression for $r$ into the centripetal acceleration formula:
$a=\frac{v^2}{\left(\frac{vT}{2\pi }\right)}$
Simplifying the expression by multiplying by the reciprocal of the denominator:
$a=v^2\cdot \frac{2\pi }{vT}$
One factor of $v$ cancels out from the numerator and denominator:
$a=\frac{2\pi v}{T}$

Thus, the acceleration of the particle is $\frac{2\pi v}{T}$.