Problem 30 - Master Resource Book in JEE Main Physics

Question

The kinetic energy \(k\) of a particle moving along a circle of radius \(R\) depends upon the distance \(s\) as \(k=a s^{2}\). The force acting on the particle is (a) \(2 a \frac{s^{2}}{R}\) (b) \(2 a s\left[1+\frac{s^{2}}{R^{2}}\right]^{t / 2}\) (c) \(2 a\) (d) \(2 \underline{a}\)

Step-by-Step Solution

Answer

The force acting on the particle is \( 2a\frac{s^2}{R} \). Choice (a) is correct.

Step 1: Understand Kinetic Energy

The given kinetic energy is defined by the expression \( k = a s^2 \), where \( a \) is a constant, and \( s \) is the distance traveled by the particle along the circle. This expression tells us how the kinetic energy depends on the distance \( s \).

Step 2: Use the Kinetic Energy Formula

We need to recall that the kinetic energy of a particle moving with velocity \( v \) is also given by \( k = \frac{1}{2}mv^2 \), where \( m \) is the mass of the particle. Since \( k = a s^2 \), we equate this to \( \frac{1}{2}mv^2 \). Thus, \( \frac{1}{2}mv^2 = a s^2 \) which simplifies to \( v^2 = \frac{2as^2}{m} \).

Step 3: Determine the Velocity Relationship

Assuming the particle travels with a uniform speed, centripetal acceleration is given by \( a_c = \frac{v^2}{R} \), where \( R \) is the radius of the circle. Substitute \( v^2 = \frac{2as^2}{m} \) from the previous step, giving \( a_c = \frac{2as^2}{mR} \).

Step 4: Relate Acceleration to Force

Apply Newton's second law, which states \( F = ma \), where \( F \) is the force and \( a \) is the acceleration. Since the acceleration \( a_c = \frac{2as^2}{mR} \), the force can be calculated as \( F = m \cdot \frac{2as^2}{mR} = \frac{2as^2}{R} \). Thus, the force \( F = 2a\frac{s^2}{R} \).

Step 5: Identify Correct Option

The result from Step 4 matches with option (a) from the given statements: \( 2a\frac{s^2}{R} \). Thus, the correct answer is option (a).

Key Concepts

Centripetal Force

Centripetal force plays a crucial role when an object moves in a circular path. It is the force that keeps the object moving in its circular trajectory by pointing towards the center of the circle. Without this force, the object would continue in a straight line due to its inertia. The centripetal force can be calculated by the formula:\[F_c = rac{mv^2}{R}\]where:- \( F_c \) is the centripetal force,- \( m \) is the mass,- \( v \) is the velocity,- \( R \) is the radius of the circle.In the original exercise, the centripetal acceleration is determined to be \( a_c = \frac{2as^2}{mR} \), which is derived from the kinetic energy equation and velocity relationship. Thus, the centripetal force for this situation is computed as \( F = 2a\frac{s^2}{R} \). This matches the given option (a) for the force acting on the particle. Understanding the centripetal force is essential because it explains how objects are able to remain in circular motion.

Uniform Circular Motion

Uniform circular motion refers to the movement of an object along a circular path with a constant speed. Even though the speed is constant, the object is continuously changing direction, which means there is always an acceleration present. This is called centripetal acceleration and is essential to maintaining the object's circular path:\[a_c = \frac{v^2}{R}\]where:- \( a_c \) is the centripetal acceleration,- \( v \) is the velocity,- \( R \) is the radius of the circle.In the problem at hand, the motion is described by the kinetic energy relation \( k = a s^2 \), and through various steps, it shows how the centripetal acceleration \( a_c \) is calculated. Uniform circular motion highlights that despite unchanging speed, continuous directional change is induced by this constant centripetal force.

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that describes the relationship between the force applied on an object, its mass, and the acceleration it experiences. It is articulated through the formula:\[F = ma\]where:- \( F \) is the force applied,- \( m \) is the mass,- \( a \) is the acceleration.In the exercise provided, this law is utilized to relate the centripetal acceleration to the force experienced by a particle in circular motion. After calculating the centripetal acceleration \( a_c = \frac{2as^2}{mR} \), the exercise applies Newton's Second Law to find that \( F = m \cdot a_c \) results in \( F = 2a\frac{s^2}{R} \), which matches option (a). Understanding this law helps clarify how forces work in tandem with mass and acceleration to dictate motion, underscoring the importance of this law in analyzing physical phenomena.