Question

A pendulum bob of mass $m$ connected to the end of an ideal string of length $l$ is released from horizontal position as shown in the given figure. At the lowest point, the bob makes an elastic collison with a stationary block of mass $5m$, which is kept on a frictionless surface. Choose the correct statement(s).

• A. Tension in the string just after impact is $\frac{17}{9}mg$.
• B. Tension in the string just after impact is $mg$.
• C. The velocity of the block just after impact is $\sqrt{\frac{2gl}{3}}$.
• D. The maximum height attained by the pendulum bob after impact is $\frac{4l}{9}$ (measured from the lowest position)

Answer 1

Answer: (A), (D)

The maximum height attained by the pendulum bob after impact is $\frac{4l}{9}$ (measured from the lowest position)

Using conservation of mechanical energy,
$mgl=\frac{1}{2}m{v}^2$
So, the velocity of the bob just before the impcat is $v=\sqrt{2gl}$ along the horizontal direction.

Let the velocity of the bob and block just after the impact be $v_1$ and $v_2$ respectively.

Since, the tension here does not provide any impluse, the momentum of the system will be conserved just before and just after the collision.


Hence, from the momentum conservation,
$mv=-m{v}_1+5m{v}_2$
$\Rightarrow 5v_2-v_1=v....\left(1\right)$

Since the collision is elastic, coefficient of restitution $e=1$,
$\Rightarrow 1=\frac{v_1+v_2}{v}$
$\Rightarrow v_1+v_2=v....\left(2\right)$

On solving (1) and (2), we get
$v_1=\frac{2v}{3}=\frac{2}{3}\sqrt{2gl}$
&
$v_2=\frac{v}{3}=\frac{1}{3}\sqrt{2gl}$

For tension in string:
Since the particle is moving in a circle, immediately after the impact, we can write
$T-mg=\frac{m{v}_1^2}{l}$

On substituting the value of $v_1$,

$\Rightarrow T-mg=\frac{m{\left(\frac{2\sqrt{2gl}}{3}\right)}^2}{l}$

$\Rightarrow T=\frac{17}{9}mg$

Let maximum height attained by the bob be $h$. Then, by conservation of mechanical energy,
$\frac{m{v}_1^2}{2}=mgh$

$\Rightarrow h=\frac{{\left(\frac{2\sqrt{2gl}}{3}\right)}^2}{2g}$

$\Rightarrow h=\frac{4l}{9}$

Hence, correct options are (a) and (d).