Question

A simple pendulum is oscillating between extreme positions $P$ and $Q$ about the mean position $O$. Which of the the following statements are true about the motion of pendulum?

• A. At point $O$, the acceleration of the bob is difference from zero.
• B. The acceleration of the bob is constant throughout the oscillation
• C. The tension in the string is constant throughout the oscillation
• D. The tension is maximum at $O$ and minimum at $P$ or $Q$

Answer 1

Answer: (A), (D)

The correct options are
A At point $O$, the acceleration of the bob is difference from zero.
D The tension is maximum at $O$ and minimum at $P$ or $Q$

Statement (a) is correct. At any position $O$ and $P$ or between $O$ and $Q$, there are two accelerations --- a tangential acceleration $g\sin \alpha$ and a centripetal acceleration $v^2/l$ (because the pendulum moves along the arc of a circle or radius $l$), where $l$ is the length of the pendulum and $v$ its speed at that position. When the bob is at the mean position $O$, the angle $\alpha =0$, therefore $\sin \alpha =0$; hence, the tangential acceleration is zero. But at $O$, speed $v$ is maximum and the centripetal acceleration $v^2/l$ is directed radially towards the point of support. When the bob is at the end points $P$ and $Q$ the speed $v$ is zero, hence the centripetal acceleration is zero at the end, points, but the tangential acceleration is maximum and is directed along the tangent to the curve at $P$ and $Q$. The tension in the string is not constant throughout the oscillation. At any position between $O$ and end point $P$ or $Q$, the tension in the string is given by $T=mg\cos \alpha$.
At the end point $P$ and $Q$ the value of $\alpha$ is the greatest, hence the tension is the least. At the mean position $O$. $\alpha =0$ and $\alpha =1$ which is the greatest; hence tension is greatest at the mean position.