Question

A simple pendulum is suspended from the ceiling of a lift. When the lift is at rest its time period is $T$. With what acceleration should the lift be accelerated upwards in order to reduce its period to $\frac{T}{2}$? ($g$ is acceleration due to gravity).

• A. $1g$
  
• B. $2g$
  
• C. $3g$
  
• D. $4g$
  

Answer 1

The correct option is C $3g$

When lift moves in upward direction with an acceleration, pendulum will be acted upon by a pseudo acceleration same to the acceleration of lift. this acceleration will be in downward direction on pendulum.

Now, time period of pendulum,

$T\propto \frac{1}{\sqrt{g}}$

If $T^{\prime }$ be the period $g^{\prime }$ be the acceleration on pendulum when lift is in motion.

$\frac{T^{\prime }}{T}=\sqrt{\frac{g}{g^{\prime }}}$

Given that $T^{\prime }=\frac{T}{2}$

So, $\frac{1}{2}=\sqrt{\frac{g}{g^{\prime }}}$

$\Rightarrow g^{\prime }=4g$

Again, $g^{\prime }=g+a=4g$

$\Rightarrow a=3g$