Question

The time period of a bar pendulum when suspended at distances 30 cm and 50 cm from its centre of gravity comes out to be the same. If the mass of the body is 2kg. Find out its moment of inertia about an axis passing through the first point.
A) 0.24 kg \[{{m}^{2}}\]
B) 0.72 kg \[{{m}^{2}}\]
C) 0.48 kg \[{{m}^{2}}\]
D) data insufficient

Answer 1

Hint: We are given a bar pendulum and when it is suspended at two lengths, the centre of gravity comes out to be the same. Given the mass is 2 kg. Angular oscillations of rigid bodies in the form of a long bar can be used to find acceleration due to gravity.

Complete step by step answer:
The bar pendulum is free to oscillate about the knife-edge as an axis. The restoring torque τ for an angular displacement θ is given by,
\[\tau =mgl\sin \theta \]
Since the angle is very small, it can be written as \[\tau =mgl\theta \]
Substituting the values,
For first case: \[{{\tau }_{1}}=mg\times 0.3\times \theta \]
For second case: \[{{\tau }_{2}}=mg\times 0.5\times \theta \]
Let the time period of the pendulum be T, by using the parallel axis theorem,
Moment of inertia is
$
{{I}_{1}}=I+2\times {{0.3}^{2}} \\
\implies {{I}_{2}}=I+2\times {{0.5}^{2}} \\
$
Because time period of blue points is equal,
$
{{I}_{1}}=I+2\times {{0.3}^{2}} \\
\implies {{I}_{2}}=I+2\times {{0.5}^{2}} \\
\implies 2\pi \sqrt{\dfrac{{{I}_{1}}}{{{\tau }_{1}}}}=2\pi \sqrt{\dfrac{{{I}_{2}}}{{{\tau }_{2}}}} \\
\implies \dfrac{{{I}_{1}}}{{{\tau }_{1}}}=\dfrac{{{I}_{2}}}{{{\tau }_{2}}} \\
$
Putting out the values of the moment of inertia,
$
\dfrac{I+2\times {{0.3}^{2}}}{I+2\times {{0.5}^{2}}}=\dfrac{0.3mg}{0.5mg} \\
\implies \dfrac{I+2\times {{0.3}^{2}}}{I+2\times {{0.5}^{2}}}=\dfrac{3}{5} \\
\implies I=0.30kg{{m}^{2}} \\
$
So, the value of moment of inertia comes out to be \[0.30kg{{m}^{2}}\]
moment of inertia about an axis passing through first point can be find as
$
{{I}_{1}}=0.3+2\times {{0.3}^{2}} \\
 =0.48kg{{m}^{2}} \\
$

So, the correct answer is “Option C”.

Note:
An ideal simple pendulum cannot be realized under laboratory conditions. Hence, a compound pendulum is used to determine the acceleration due to gravity in the laboratory. It is widely used and an accurate experiment as direct measurement of the acceleration due to gravity is very difficult.