Which of the following is NOT an example of inertia at rest?

The correct option is The sudden application of brakes in a vehicle.

Inertia of Rest is the inherent property of a body by virtue of which it resists any change in its state of rest unless acted upon by an external unbalanced force. That is, a body initially at rest tends to remain at rest.

Let us analyze each of the given options:
1. Falling down of dust particles while dusting a cloth: When a cloth is suddenly beaten or dusted, the cloth moves, but the dust particles tend to stay at rest due to inertia of rest and fall off.
2. Falling down backward when a vehicle starts immediately: When a stationary vehicle suddenly accelerates forward, the passenger's lower body moves with the vehicle while the upper body tends to remain at rest due to inertia of rest, causing the passenger to fall backward.
3. The detachment of leaves/fruits due to shaking: Shaking a tree branch sets the branch in motion, while the attached leaves and fruits tend to remain at rest due to inertia of rest, causing them to detach and fall.

4. The sudden application of brakes in a vehicle: When brakes are applied suddenly to a moving vehicle, the vehicle and the lower part of a passenger's body come to a stop. However, the upper part of the body continues moving forward due to inertia of motion, causing the passenger to lurch forward. Therefore, this is an example of inertia of motion, NOT inertia at rest.

Hence, the sudden application of brakes in a vehicle is NOT an example of inertia at rest.