Animated Solution for Physics - Laws of Motion: A lift is moving down with acceleration $a$. A man in the lift drops a ball inside the lift. The acceleration of the ball as observed by the man in the lift and a man standing stationary on the ground are respectively

Unmasking Pseudo Forces

Imagine you are standing inside an elevator that is accelerating downwards, and you drop a ball. At the exact same moment, your friend is watching this happen from the ground outside through a glass window.

Will both of you measure the same acceleration for the falling ball?

This classic physics problem forces us to confront the reality of frames of reference and the illusion of pseudo forces. Let's break it down by looking through the eyes of both observers.

Let's first talk about your friend standing on the ground. The ground is what physicists call an inertial frame of reference—it is not accelerating.

When you drop the ball, it is no longer physically connected to the elevator. From the perspective of the ground observer, the ball is simply in free fall. The only real force acting on it is the Earth's gravity pulling it down.

Therefore, the net force is simply $F_{net}=mg$.

Using Newton's second law, the acceleration of the ball as observed from the ground is:

It doesn't matter what the elevator is doing; for the outside observer, a dropped object falls with an acceleration of $g$.

Now, let's shift our focus to you, standing inside the elevator. Because the elevator is accelerating downwards with an acceleration $a$, it is a non-inertial frame.

If you try to apply Newton's laws directly in an accelerating frame, they seem to break. To fix this and make the math work, we must introduce a fictitious force, known as a pseudo force.

The golden rule for a pseudo force is simple: it always acts in the direction exactly opposite to the acceleration of the frame. Since the elevator is accelerating downwards with $a$, the pseudo force on the ball will act upwards with a magnitude of $ma$.

So, in your frame of reference inside the lift, there are two competing forces acting on the ball: 1. Gravity is pulling it down with $mg$. 2. The pseudo force is pushing it up with $ma$.

The net downward force is therefore:

To find the apparent acceleration of the ball as seen by you, we divide this net force by the mass of the ball:

We have our answers! The man inside the lift sees the ball falling with an acceleration of $g-a$, while the man on the ground sees it falling with an acceleration of $g$.

Food for thought: What if the elevator cable snaps and it goes into free fall? In that case, the elevator's acceleration $a$ becomes equal to $g$. The apparent acceleration $g-a$ would become zero! The pseudo force would perfectly cancel gravity, and the ball would just float in front of you. Fascinating, right?