Weigh of a moving body in a convex bridge is Less than the weight of the same car resting on the same bridge, explain why?

When a car moves over a convex bridge, its weight appears to be less than when it is stationary on the same bridge. This phenomenon can be explained through the concepts of forces, acceleration, and the shape of the bridge. Let’s break it down step by step.

The Role of Forces

To understand why the weight of a moving car seems to decrease, we first need to clarify what we mean by "weight." The weight of an object is the force exerted on it due to gravity, which remains constant regardless of whether the object is moving or stationary. However, when the car is in motion, particularly on a convex surface, the forces acting on it change.

Understanding the Forces at Play

When the car is at rest on the bridge, the only forces acting on it are its weight (downward) and the normal force from the bridge (upward). These two forces are equal, resulting in a net force of zero, which keeps the car stationary.

However, when the car moves over the convex bridge, it experiences a change in motion. The bridge's curvature causes the car to follow a circular path. As the car moves upward along the curve, it accelerates, and this acceleration affects the forces acting on it.

Acceleration and Apparent Weight

As the car ascends the convex part of the bridge, it is not only subject to gravitational force but also to centripetal acceleration, which is directed towards the center of the circular path. This acceleration effectively reduces the normal force exerted by the bridge on the car. The normal force is what we perceive as the "weight" of the car when it is on the bridge.

• Weight (W): The gravitational force acting on the car, which is constant.
• Normal Force (N): The force exerted by the bridge on the car, which changes when the car is in motion.
• Centripetal Acceleration (a): The acceleration required to keep the car moving in a circular path, which affects the normal force.

The relationship can be expressed as:

N = W - ma

Where:

• N = normal force (apparent weight)
• W = weight of the car
• m = mass of the car
• a = centripetal acceleration

Example in Action

Imagine a car weighing 1,000 kg moving over a convex bridge. The gravitational force acting on it is:

W = mg = 1,000 kg × 9.81 m/s² = 9,810 N

As the car reaches the highest point of the bridge, let’s say it experiences a centripetal acceleration of 2 m/s². The normal force can be calculated as:

N = W - ma = 9,810 N - (1,000 kg × 2 m/s²) = 9,810 N - 2,000 N = 7,810 N

In this scenario, the car's apparent weight while moving over the bridge is 7,810 N, which is less than its actual weight of 9,810 N when stationary.

Conclusion

This reduction in the normal force, or apparent weight, occurs because the car is not only being pulled down by gravity but also needs to maintain its circular motion, which requires an upward force that is partially provided by the bridge. Thus, while the car is in motion on a convex bridge, it feels lighter due to the dynamics of acceleration and the forces involved.