Problem 19 - Master Resource Book in JEE Main Physics

Question

A body of mass \(m\) is moving in a circle of radius \(r\) with a constant speed \(v\). The force on the body is \(m v^{2} / r\) and is directed towards the centre. What is the work done by this force in moving the body over half the circumference of the circle? (a) \(\frac{m v^{2}}{r} \times \pi r\) (b) \(\frac{m v^{2}}{r^{2}}\) (c) zero (d) \(\frac{\pi r}{m v^{2}}\)

Step-by-Step Solution

Answer

The work done is zero, as the force is perpendicular to the displacement.

Step 1: Understanding the Work Done

Work is defined as the component of force in the direction of displacement multiplied by the displacement. Mathematically, work is given by the formula: \[ W = F \cdot d \cdot \cos \theta \]where \( F \) is the magnitude of the force, \( d \) is the magnitude of displacement, and \( \theta \) is the angle between the force and displacement vectors.

Step 2: Analyzing the Force Direction

The force on the body moving in a circle is centripetal, pointing towards the center. The displacement over half the circumference of the circle is along the path of the circle, tangent to it at every point. This means the angle \( \theta \) between the centripetal force and the displacement along the circumference is \( 90^\circ \).

Step 3: Calculating Cosine of the Angle

Calculate \( \cos \theta \) for the angle \( \theta = 90^\circ \):\[ \cos 90^\circ = 0 \]

Step 4: Determining the Work Done

Since the force is perpendicular to the displacement at every point:\[ W = F \cdot d \cdot \cos 90^\circ = F \cdot d \cdot 0 = 0 \]This implies that no work is done by the centripetal force as there is no displacement component in the direction of the force.

Key Concepts

Centripetal Force

Centripetal force is a force that acts on an object moving in a circular path.
This force is always directed towards the center of the circle around which the object is moving. It's important to understand that centripetal force doesn't act to change the object's speed but rather keeps it in circular motion.
For an object of mass \( m \), moving at a constant speed \( v \) along a circular path of radius \( r \), the magnitude of the centripetal force \( F_c \) can be calculated using:

• \( F_c = \frac{m v^2}{r} \)

This formula highlights a critical characteristic of circular motion: the centripetal force is dependent on the square of the velocity and inversely proportional to the radius of the circle.
This means that the force becomes significant if the object moves faster or if the circle's radius is smaller.

Circular Motion

Circular motion refers to movement along a circular path and involves a constant change in direction, even if the speed is constant. One key feature of circular motion is that although the speed can be constant, the velocity is not because velocity includes direction.
In uniform circular motion, the speed remains constant and the path is a perfect circle.
Let's explore some key points about circular motion:

• The radius \( r \) is the fixed distance from the center of the circle to any point on its circumference.
• The circumference of a circle is \( 2 \pi r \), which is the total distance around the circle.
• As a body travels along this path, its direction changes continuously, requiring a centripetal force to maintain this motion.

Understanding the characteristics of circular motion helps explain why centripetal forces do no work, as the force acts perpendicular to the direction of motion at every point.

Work Energy Theorem

The Work Energy Theorem connects the concepts of work and energy, stating that the work done by all forces acting on an object leads to a change in the kinetic energy of the object.
Mathematically, it is expressed as:

• \( W = \Delta K \)

Where \( W \) is the work done, and \( \Delta K \) is the change in kinetic energy.
In the context of circular motion and centripetal force, the work-energy theorem has an interesting outcome. Although an object moves under the influence of a centripetal force, the work done is zero. This is because the centripetal force acts perpendicular to the displacement.
Due to this perpendicular action, there is no work done in the direction of force, and thus any kinetic energy related to the speed of the object remains unchanged throughout the motion. This highlights that while energy is required to maintain the centripetal force, it does not add or remove energy from the system.