Animated Solution for Physics - Work, Energy, and Power: A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a_c$ is varying with time $t$ as $a_c=k^{2}r{t}^2$, where $k$ is a constant. The power delivered to the particle by the force acting on it is

The beauty of circular motion lies not just in the fact that an object is going in circles, but in the intricate dance of forces that keep it there and dictate its speed. When a particle moves in a circular path, it experiences a fascinating interplay of kinematics and dynamics. Let's dive deep into this classic JEE problem and unravel the physics behind it.

A Particle in a Circle

Imagine a particle of mass $m$ constrained to move in a circular path of a constant radius $r$. We are given a very specific condition: its centripetal acceleration $a_c$ is not constant, but varies with time $t$ according to the equation:

Here, $k$ is just a positive constant. What does this equation physically mean? Centripetal acceleration is the inward "pull" required to keep the particle turning. If this pull is increasing quadratically with time, it implies the particle must be spinning faster and faster. It's like driving a car in a circle and continuously pressing the accelerator pedal.

To understand how fast the particle is moving, we need to connect the given centripetal acceleration to its speed. The fundamental kinematic relationship for circular motion tells us that centripetal acceleration is the square of the speed divided by the radius:

By equating our theoretical formula with the given expression, we can unlock the velocity of the particle:

Multiplying both sides by $r$, we isolate $v^2$:

Taking the positive square root (since speed is the magnitude of velocity and must be non-negative), we find:

This is a crucial breakthrough! The speed of the particle is increasing linearly with time.

In circular motion, acceleration has two distinct jobs. The centripetal acceleration ($a_c$) changes the direction of the velocity vector, keeping the particle on the circular track. However, because the speed $v$ is changing, there must be another component of acceleration responsible for this change in magnitude. This is the tangential acceleration ($a_t$).

By definition, tangential acceleration is the rate of change of speed:

Substituting our expression for $v$:

Since $k$ and $r$ are constants, the tangential acceleration is constant! Now, according to Newton's Second Law, a tangential acceleration requires a tangential force ($F_t$).

So, a constant tangential force is continuously pushing the particle along the circular track, causing its speed to increase linearly.

The ultimate goal of the problem is to find the power delivered to the particle. Power is defined as the rate at which work is done, which can be elegantly expressed as the dot product of the net force vector and the velocity vector:

The net force on the particle has two components: the centripetal force (${\vec{F}}_c$) and the tangential force (${\vec{F}}_t$).

Here is the catch: The centripetal force always points radially inward, exactly perpendicular ($90^{\circ }$) to the instantaneous velocity vector. Because the dot product of perpendicular vectors is zero ($\cos 90^{\circ }=0$), the centripetal force does zero work and delivers zero power. It merely steers the particle.

Therefore, all the power is delivered exclusively by the tangential force, which is perfectly parallel to the velocity:

Substituting the values we derived earlier:

Work-Energy Theorem

While the force method is highly instructive, there is a much faster, more elegant way to solve this using the Work-Energy Theorem. The theorem states that the net work done on a particle equals its change in kinetic energy. Consequently, Power (the rate of doing work) is simply the rate of change of kinetic energy!

First, write down the kinetic energy ($K$) using the speed we found ($v=krt$):

Now, just differentiate this kinetic energy with respect to time to get the power:

In just two lines of calculus, we arrive at the exact same result! This highlights the profound consistency and beauty of physics. Whether you analyze the forces or track the energy, the universe always balances its books.