Problem 5 - Master Resource Book in JEE Main Physics

Question

The kinetic energy \(K\) of a particle moving along a circle of radius \(R\) depends on the distance covered \(s\) as \(K=a s^{2}\), where \(a\) is a constant. The force acting on the particle is (a) \(2 a \frac{5^{2}}{R}\) (b) \(2 \operatorname{as}\left(1+\frac{s^{2}}{R^{2}}\right)^{1 / 2}\) (c) 2 as (d) \(2 a \frac{R^{2}}{s}\)

Step-by-Step Solution

Answer

Option (c): \(2as\) is the correct answer.

Step 1: Understand the Problem

We are given that the kinetic energy of a particle is described by the formula \( K = a s^2 \). We need to calculate the force acting on the particle as it moves along the circle.

Step 2: Recall Kinetic Energy and Force Relationship

Force is related to the change in kinetic energy by the work-energy principle. In translational motion, force can also be related to the derivative of kinetic energy with respect to the distance moved, which gives the force acting in the direction of motion.

Step 3: Calculate the Force

The force exerted on the particle can be derived from the derivative of kinetic energy \( K \) with respect to distance \( s \):\[F = \frac{dK}{ds} = \frac{d}{ds}(as^2) = 2as\]This force, \( F = 2as \), is in the tangential direction along the path.

Step 4: Select the Correct Answer

The force we calculated \( F = 2as \) matches option (c) in the given choices. Thus, the correct answer is option (c).

Key Concepts

Work-Energy Principle

The work-energy principle is a fundamental concept in physics that links the idea of work done by forces with changes in kinetic energy. Essentially, this principle states that the work done on an object is equal to the change in its kinetic energy. In formula terms, if a force moves an object along a path, the work-energy principle can be expressed as:

• Work done = Change in Kinetic Energy

This means if a particle, like in our problem, moves in a circular path and its kinetic energy changes, the work done by the force on this particle causes that change.
It's an incredibly useful principle because it provides a straightforward way to calculate the effect of forces without delving into the forces at every point along the way. It transforms the problem of understanding force interactions into an analysis of energy changes. The work-energy principle simplifies problem-solving when dealing with dynamic systems in motion.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. For any moving particle, calculating the kinetic energy involves its mass and its speed or velocity. The general formula is:

• \[ K = \frac{1}{2} m v^2 \]where \( K \) is the kinetic energy, \( m \) is the mass, and \( v \) is the speed or velocity.

In the context of our specific problem, we're given that the kinetic energy is not dependent on speed directly, but rather on the square of the distance \( s \) the particle has traveled, making the formula:

• \[ K = a s^2 \]

Here, \( a \) is a constant, and \( s \) is the distance covered.
This relationship indicates that as the particle moves a greater distance along its circular path, its kinetic energy increases in proportion to the square of that distance. Such dependency suggests a specific kind of force at play, influencing how energy changes as movement occurs.

Force Derivation

In physics, deriving the force from a given energy equation involves examining how the kinetic energy changes with respect to distance. For a particle moving along a circular path, the kinetic energy equation is given as \( K = a s^2 \). To find the force acting on the particle, we utilize the relationship between force and energy changes.The force can be obtained by taking the derivative of the kinetic energy with respect to distance, \( s \):

• \[ F = \frac{dK}{ds} = \frac{d}{ds}(a s^2) \]
• Evaluating this derivative gives us:\[ F = 2as \]This result shows the force depends linearly on the distance \( s \).

This force acts tangentially to the direction of motion, which is typical for circular motion since the particle's velocity vector is always tangent to the path.
Understanding how to derive this force helps us make sense of how the object interacts with its surroundings, reaffirming the principles of energy transformations and motion. This derivation is not only illustrative of the concepts but also confirms the physical reality of the motion described.