Question

A block of mass $M$ rests on a smooth horizontal table. There is a small gap in the table under the block through which a pendulum has been attached to the block. The bob of the simple pendulum has mass $m$ and length of the pendulum is $L$. The pendulum is set into small oscillations in the vertical plane of the figure. Calculate its time period. The table does not interfere with the motion of the string.

Answer 1

$2\pi\sqrt{\frac{LM}{g(M+m)}}$

Answer 2

1. Define the generalized coordinates for the system: the horizontal displacement of the block ($X$) and the angular displacement of the pendulum ($\theta$).

2. Calculate the kinetic energy of the block and the bob in terms of these coordinates and their time derivatives. Use the small angle approximation for the bob's velocity components.

3. Calculate the potential energy of the system due to gravity acting on the bob, using the small angle approximation for the height.

4. Formulate the Lagrangian $L = T - V$.

5. Apply the Euler-Lagrange equations for the generalized coordinates $X$ and $\theta$ to obtain the equations of motion.

6. Solve the coupled differential equations to find the equation of motion for $\theta$. This results in a second-order linear differential equation in $\theta$, which is the equation of a simple harmonic oscillator.

7. From the equation of simple harmonic motion, identify the angular frequency $\omega$ and calculate the time period $T = \frac{2 \pi}{\omega}$.

Answer: $2\pi\sqrt{\frac{LM}{g(M+m)}}$