A particle of mass m moves along a horizontal circle of radius R such that normal acceleration of particle varies with time as `a_(n)=kt^(2).` where k is a constant.
Power developed by total at time T is

Options

A. `(mkRT)/(3)`
B. `2mkRT`
C. `(mkRT)/(2)`
D. mkRT

Answer

To solve the problem step by step, we need to find the power developed by the particle at time \( T \) given that its normal acceleration varies with time as \( a_n = kt^2 \).

### Step 1: Understand the relationship between normal acceleration and velocity
The normal acceleration \( a_n \) in circular motion is given by the formula:
\[ a_n = \frac{v^2}{R} \]
where \( v \) is the tangential velocity and \( R \) is the radius of the circular path.

### Step 2: Set up the equation using the given normal acceleration
From the problem, we know:
\[ a_n = kt^2 \]
Equating the two expressions for normal acceleration, we have:
\[ \frac{v^2}{R} = kt^2 \]

### Step 3: Solve for the tangential velocity \( v \)
Rearranging the equation gives:
\[ v^2 = kt^2 R \]
Taking the square root of both sides, we find:
\[ v = \sqrt{kt^2 R} = t\sqrt{kR} \]

### Step 4: Find the tangential acceleration \( a_t \)
The tangential acceleration \( a_t \) can be found from the relationship of normal acceleration. Since \( a_n = \frac{v^2}{R} \), we can differentiate \( v \) with respect to time to find \( a_t \):
\[ a_t = \frac{dv}{dt} = \frac{d}{dt}(t\sqrt{kR}) = \sqrt{kR} \]

### Step 5: Calculate the tangential force \( F_T \)
The tangential force \( F_T \) acting on the particle is given by:
\[ F_T = m a_t = m \sqrt{kR} \]

### Step 6: Calculate the power \( P \)
Power \( P \) is defined as the product of the tangential force and the tangential velocity:
\[ P = F_T \cdot v \]
Substituting the expressions for \( F_T \) and \( v \):
\[ P = (m \sqrt{kR}) \cdot (t\sqrt{kR}) = m k R t \]

### Step 7: Final expression for power at time \( T \)
Thus, the power developed by the particle at time \( T \) is:
\[ P = m k R T \]

### Summary
The power developed by the particle at time \( T \) is given by:
\[ \boxed{m k R T} \]