Question

Give an example where (a) the velocity of a particle is zero but its acceleration is not zero, (b) the velocity is opposite in direction to the acceleration, (c) the velocity is perpendicular to the acceleration.

Answer 1

For the velocity to be zero and the acceleration to be non zero, there must be a net force on the particle even when it is at rest. The direction of the acceleration is decided by the direction of the net force, not that of the velocity.

Complete solution:
(a). In the motion of a simple pendulum, when the bob of the pendulum is at either one of the two extreme positions, its potential energy is maximum and its kinetic energy is equal to zero. Zero kinetic energy means zero-velocity of the bob at the extreme position. But we know that the bob is continuously acted upon by the gravitational force, acting along the tangent to the circular path of the bob, and equal to $mg\sin {\theta }$, where $m$ is the mass of the bob and ${\theta }$ is the angle made by the string of the pendulum with the vertical. So the acceleration of the bob at this instant is equal to $g\sin {\theta }$.
(b). When a body is thrown vertically upwards its velocity is in the vertically upward direction. But as we know that the gravitational force always acts in the vertically downward direction, so the acceleration of the body is in a vertically downward direction. Hence, the velocity of the body is opposite to its acceleration.
(c). We know that a particle is able to move in a uniform circular motion due to the centripetal force acting on it towards the centre of the circle. So this means that the acceleration of a body moving in the uniform circular motion is towards the centre. But since the body is moving along the circle, so the velocity of the body is along the tangent to the circle. We know that the radius is perpendicular to the tangent, so in this case, the velocity of the body is perpendicular to its acceleration.

Note: Do not think that in any circular motion, the velocity of the particle will be perpendicular to its acceleration. This statement is true only for the uniform circular motion, where the tangential acceleration of the particle is equal to zero. But in a non-uniform circular motion, the particle has both tangential and centripetal accelerations, due to which its net acceleration is not towards the centre and hence is not perpendicular to the velocity, which is always along the tangent to the circle.