Problem 28 - Master Resource Book in JEE Main Physics

Question

A body of mass \(1 \mathrm{~kg}\) is rotating in a vertical circle of radius \(1 \mathrm{~m}\). What will be the difference in its kinetic energy at the top and bottom of the circle? (Take \(g=10 \mathrm{~ms}^{-2}\) ) (a) \(10 \mathrm{~J}\) (b) \(20 \mathrm{~J}\) (c) \(30 \mathrm{~J}\) (d) \(50 \mathrm{~J}\)

Step-by-Step Solution

Answer

(b) 20 J

Step 1: Understand the problem

We need to find the difference in kinetic energy when the body is at the top and the bottom of the vertical circle. Given that the mass of the body is \(1 \, \text{kg}\), the radius of the circle is \(1 \, \text{m}\), and the acceleration due to gravity \(g\) is \(10 \, \text{ms}^{-2}\).

Step 2: Calculate the Potential Energy Difference

At the top of the circle, the height is equal to 2 times the radius (2 meters) from the lowest point. Thus, the potential energy at the top is \( mg \times 2 \) where \( m = 1 \text{ kg} \) and \( g = 10 \, \text{ms}^{-2} \). Therefore, the change in potential energy \( \Delta U = 1 \times 10 \times 2 = 20 \, \text{J}\).

Step 3: Use Energy Conservation Principle

According to the conservation of mechanical energy, the change in potential energy will be equal to the change in kinetic energy, assuming no non-conservative forces do work. Therefore, the change in kinetic energy \( \Delta K = \Delta U = 20 \, \text{J} \).

Step 4: Conclusion

The kinetic energy is higher at the bottom than at the top due to the energy conservation. Since the potential energy difference is \(20 \, \text{J}\), the difference in kinetic energy from top to bottom is also \(20 \, \text{J}\).

Key Concepts

Vertical Circle

A vertical circle is a path along which an object moves in a circle that is oriented perpendicular to the ground. This type of motion is quite common in physics and mechanical applications, such as in a roller coaster or a pendulum motion. When an object follows a vertical circular path, it experiences changes in both its potential energy and kinetic energy as it moves from the top to the bottom of the circle and vice versa. Because gravity acts downward, it influences the net force acting on the object at various points of the circle. This interaction between gravity and the object's motion has important implications for calculating the energies involved in the motion.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. It is calculated using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object. In the context of a vertical circle, the kinetic energy changes as the object moves along different heights. At the top of the circle, the object has lower kinetic energy because it has converted much of its energy into potential energy. As the object drops toward the bottom of the circle, its velocity increases, consequently increasing its kinetic energy due to the conversion of potential energy back to kinetic energy.

Potential Energy

Potential energy in a vertical circle is primarily gravitational potential energy, which depends on the height above a reference point. It is calculated as \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height. When an object is at the top of the vertical circle, it has maximum potential energy, given that it is at the highest point. At the lowest point, the potential energy is minimum because the height is zero. The change in potential energy, as derived in the task, indicates how much potential energy has been converted into kinetic energy (or vice versa) as the body moves across different points of the circle.

Energy Conservation

The principle of energy conservation states that in a closed system, the total energy remains constant, though it may change forms. In the context of a vertical circle, the object’s mechanical energy, which is the sum of kinetic and potential energy, remains unchanged unless external work is done. As the object moves from the top of the circle to the bottom, gravitational forces conservatively convert potential energy into kinetic energy. This allows us to state that the difference in potential energy caused by a change in height equals the difference in kinetic energy. In our specific scenario, the potential energy difference was calculated as 20 J, and thus, by energy conservation, the kinetic energy difference is also 20 J.

Circular Motion

Circular motion involves an object moving along a circular path. In a vertical circle, this type of motion requires a continuous change in direction, which is facilitated by a net inward force known as centripetal force. This force keeps the object moving along the curved path. The velocity of the object, and consequently its kinetic energy, varies as it moves through different points along the path of the circle due to gravitational force acting on it. At the top of the circle, the speed is lower, resulting in less kinetic energy. As the object descends and moves toward the bottom of the circle, it speeds up, increasing kinetic energy because of the conversion of potential energy to kinetic energy. Understanding circular motion is crucial for solving problems involving objects moving in vertical circles.