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 $n^{2} R t^{2}$ where n is a constant. The power delivered to the particle by the force acting on it, is:

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

$M n R^{2} t^{2}$

$M n^{2} R^{2} t$

$M n R^{2} t$

answer is B.

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Detailed Solution

Centripetal acceleration $a_{c} = n^{2} R t^{2}$

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

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

Here power is delivered by tangential force only because power by centripetal force is zero.

$[S i n c e {\vec{F}}_{e} ⊥ \vec{V}]$

$a_{t} = \frac{d v}{d t} = n R$

$P o w e r = m a_{t} v = m \times n R × n R t = M n^{2} R^{2} t$

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