A particle of mass M is moving in a circle of fixed radius R in such a way that its centripetal acceleration at time t is given by $$n^2Rt^2$$, where $$n$$ is a constant. The power delivered to the particle by the force acting on it, is:

A particle of mass $$ M $$ is moving in a circle of fixed radius $$ R $$ with centripetal acceleration given by $$ n^2 R t^2 $$, where $$ n $$ is a constant. We need to find the power delivered to the particle by the force acting on it.

First, recall that centripetal acceleration $$ a_c $$ is related to the tangential velocity $$ v $$ by the formula:

$$ a_c = \frac{v^2}{R} $$

Given that $$ a_c = n^2 R t^2 $$, we substitute:

$$ \frac{v^2}{R} = n^2 R t^2 $$

Solving for $$ v^2 $$, multiply both sides by $$ R $$:

$$ v^2 = n^2 R t^2 \times R $$

$$ v^2 = n^2 R^2 t^2 $$

Taking the square root of both sides (and considering speed as positive, so we take the positive root):

$$ v = n R t $$

Now, centripetal acceleration changes with time, which means the speed is changing. Therefore, there must be a tangential acceleration $$ a_t $$ responsible for changing the speed. Tangential acceleration is the derivative of velocity with respect to time:

$$ a_t = \frac{dv}{dt} $$

Substitute $$ v = n R t $$:

$$ a_t = \frac{d}{dt} (n R t) $$

Since $$ n $$ and $$ R $$ are constants:

$$ a_t = n R \frac{d}{dt}(t) = n R \times 1 = n R $$

The tangential force $$ F_t $$ is given by Newton's second law:

$$ F_t = M a_t = M \times n R = M n R $$

The centripetal force $$ F_c $$ is perpendicular to the velocity and does no work, so it delivers no power. Power is delivered only by the tangential force component, which is parallel to the velocity. Power $$ P $$ is the dot product of force and velocity:

$$ P = \vec{F} \cdot \vec{v} = F_t \times v \quad (\text{since they are parallel}) $$

Substitute $$ F_t = M n R $$ and $$ v = n R t $$:

$$ P = (M n R) \times (n R t) $$

$$ P = M n R \times n R t $$

$$ P = M n^2 R^2 t $$

Comparing with the options:

A. $$ \frac{1}{2} M n^2 R^2 t^2 $$

B. $$ M n^2 R^2 t $$

C. $$ M n R^2 t^2 $$

D. $$ M n R^2 t $$

Option B matches our result. Hence, the correct answer is Option B.