A particle of mass `m` starts moving in a circular path of canstant radiur `r` , such that iss centripetal acceleration`a_(c)` is varying with time a=`t` as
`(a_(c)=k^(2)r//t)` , where `K` is a contant. What is the power delivered to the particle by the force acting on it ?

Explanation

To solve the problem, we need to find the power delivered to a particle of mass \( m \) moving in a circular path of constant radius \( r \) with a centripetal acceleration that varies with time as \( a_c = \frac{k^2 r}{t} \).

### Step-by-Step Solution:

1. **Understanding Centripetal Acceleration**:
The centripetal acceleration \( a_c \) for a particle moving in a circular path is given by the formula:
\[ a_c = \frac{v^2}{r} \]
where \( v \) is the linear velocity and \( r \) is the radius of the circular path.

2. **Equating the Given Expression**:
We are given that:
\[ a_c = \frac{k^2 r}{t} \]
Setting the two expressions for centripetal acceleration equal gives:
\[ \frac{v^2}{r} = \frac{k^2 r}{t} \]

3. **Solving for Velocity \( v \)**:
Rearranging the equation to solve for \( v^2 \):
\[ v^2 = \frac{k^2 r^2}{t} \]
Taking the square root to find \( v \):
\[ v = k r \sqrt{\frac{1}{t}} = \frac{k r}{\sqrt{t}} \]

4. **Calculating Kinetic Energy \( KE \)**:
The kinetic energy \( KE \) of the particle is given by:
\[ KE = \frac{1}{2} m v^2 \]
Substituting our expression for \( v^2 \):
\[ KE = \frac{1}{2} m \left(\frac{k^2 r^2}{t}\right) = \frac{m k^2 r^2}{2t} \]

5. **Finding the Work Done \( \Delta W \)**:
The change in kinetic energy \( \Delta W \) when the particle moves from rest (initial kinetic energy = 0) to its current kinetic energy is simply:
\[ \Delta W = KE = \frac{m k^2 r^2}{2t} \]

6. **Calculating Power \( P \)**:
Power is defined as the rate of doing work, which can be expressed as:
\[ P = \frac{\Delta W}{\Delta t} \]
Since we are considering the instantaneous power, we can express it as:
\[ P = \frac{dW}{dt} \]
Substituting for \( \Delta W \):
\[ P = \frac{m k^2 r^2}{2t^2} \]

### Final Answer:
The power delivered to the particle by the force acting on it is:
\[ P = \frac{m k^2 r^2}{2t^2} \]