Question

A wagon of $m$ can move without friction along horizontal rails. A simple pendulum consisting of a sphere of mass $m$ is suspended from the ceiling of the wagon by a string of length $l$. At the initial moment the wagon and the pendulum are at rest and the string is deflected through an angle ${60}^o$ from the vertical. Find the velocity of the wagon when the pendulum passes through the mean position.

• A. $\sqrt{gl}$
• B. $\sqrt{2gl}$
• C. $\sqrt{\frac{gl}{2}}$
• D. $2\sqrt{gl}$

Answer 1

The correct option is C $\sqrt{\frac{gl}{2}}$

potential energy of pendulum is converted to kinetic energy of pendulum and kinetic energy of wagon.

assuming final velocity at mean position is $v_1$ for pendulum and $v_2$ for wagon,

so balancing the energy,

$mg(l-lcos60)=\frac{1}{2}m{v}_1^2+\frac{1}{2}m{v}_2^2$

also no external force is acting on the system in x direction so moment will be balanced in x direction.

$m{v}_1=m{v}_2$ $=>v_1=v_2=v$

we get from both the above equations

$2.\frac{1}{2}m{v}^2=mgl\frac{1}{2}$

$v=\sqrt{\frac{gl}{2}}$

so answer is C.