Question

A closed circular wire hung on a nail in a wall undergoes small oscillations of amplitude 20 and time period 2 s. Find (a) the radius of the circular wire, (b) the speed of the particle farthest away from the point of suspension as it goes through its mean position, (c) the acceleration of this particle as it goes through its mean position and (d) the acceleration of this particle when it is at an extreme position. Take g = π2 m s−2.

Answer 1

Explanation:

• Let us consider A to be the suspension point and B to be the centre of gravity.  
• We know that the moment of inertia about A is $\mathrm{I}=I_{c . g}+m h^{2}=>I=\left(\frac{m l^{2}}{3}\right)$ where l is the length of the uniform rod and $h=\frac{l}{2}$.  Thus,  $T=2 \pi \sqrt{\left(\frac{l}{m g\left(\frac{l}{2}\right)}\right)}=2 \pi \sqrt{\frac{2 l}{3 g}}$ .  
• If the time period of simple pendulum of length x is equal to the time period T,  $T=2 \pi \sqrt{\frac{x}{g}}$ where x is the length of the pendulum.
• Therefore,  $x=\frac{2 l}{3}$. Hence the length of the pendulum is  $\frac{2 l}{3}$.