Question

Prove that a motor car moving over a convex bridge is lighter than the same car resting on the same bridge.

Answer 1

Step1: When the car is stationary on the convex surface

Let's start with the part when the car is stationary on the convex surface, as depicted in the diagram below.

We can see from the diagram that the only force acting on the stationary car is gravity, which will be balanced by the normal force.
Therefore, the equation of motion can be given as $N=mg\dots \dots \dots \dots \dots \left(i\right)$
Step2: When the car is moving on the convex surface

Now, let the car be moving on the convex surface with velocity $v$ as shown in the figure below

Now, the forces on the moving car will be the force due to gravity and the force due to the centripetal acceleration caused by the velocity $v$.
Therefore, the equation of motion can be given as, $N_a=mg-\frac{m{v}^2}{r}...............\left(ii\right)$
Step3: Analyzing both the conditions

Therefore from (i) and (ii), $N>N_a$

Therefore, we can say that the moving car on the convex surface is lighter than the stationary car on the same surface.

Hence, the weight of the moving car is less than the weight of the stationary car.