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Physics Guru is an online education and competitive examinations preparation portal founded by Manish Verma, an IIT Madras alumnus, providing learning resources for competitive examinations.

The emphasis is on conceptual understanding, analytical thinking and problem solving rather than simply memorising formulas and standard methods.

The objective is to help students understand what they are learning and develop the ability to solve problems they have not seen before.

Spring + Block in Free Fall

A block of mass $m$ is suspended from the ceiling of a stationary hanging elevator through an ideal spring of spring constant $k$. Suddenly, the cable of the elevator breaks and the elevator starts falling freely. What is the min. time the block will take to attain maximum speed with respect to the elevator?

Solution

Let us place the observer inside the elevator.

Before the cable is cut, the spring is elongated.

The moment the cable is cut and the elevator falls freely, the forces acting on the block are:

$mg$ downwards

Pseudo force $ma$ upwards (here $a = g$ as this is free fall, $a$ being the acceleration of the elevator during free fall)

$kx$ upwards as the spring is elongated

So, the net force upwards = $kx$

However, this is the condition for SHM.

The time period of oscillation $T= 2\pi \sqrt \frac {m}{k}$

The speed is maximum at the mean position.

The block crosses the mean position for the first time after time $t_{min}=\frac {T}{4}=\frac {\pi}{2} \sqrt {\frac {m}{k}}$