Question

A particle of mass $2\text{kg}$ is moving in a circular path of constant radius $1\text{m}$ such that its centripetal acceleration $a_c$ is varying with time $t$ as $a_c=100t^2$. The power delivered to the particle by the force acting on it at $t=5\text{sec}$ is:

• A. $400\text{W}$
• B. $1000\text{W}$
• C. $500\text{W}$
• D. $0\text{W}$

Answer 1

The correct option is B $1000\text{W}$

Given,
Mass of the particle $\left(m\right)=2\text{kg}$
Radius of circular path $\left(r\right)=1\text{m}$
Centripetal acceleration of particle $\left(a_c\right)=100t^2$
Time $\left(t\right)=5\text{sec}$
Force acting on the particle is centripetal in nature $\left(F_c\right)$.
In circular motion, only tangential force provides power as $F_C\perp v$ and $F_t||v$


$\therefore$ $P_C=\overrightarrow{F_c}.\overrightarrow{v}=0$ and $P_t=\overrightarrow{F_t}.\overrightarrow{v}\ne 0$
Now, $a_c=100t^2$
$\Rightarrow \frac{v^2}{r}=100t^2$
$\Rightarrow$ Speed $\left(v\right)=10t$
By definition, tangential acceleration $a_t=\frac{dv}{dt}$
$\therefore a_t=10\text{m}/s^2$
i.e Tangential force $\left(F_t\right)=m{a}_t=20\text{N}$
As discussed earlier, power is delivered to the particle only by the action of tangential force.
$\therefore$ Power $\left(P_t\right)=\overrightarrow{F_t}.\overrightarrow{v}=Fv\cos \theta =20\times 10t\times \cos 0^0=200t$
At $t=5\text{sec}$
$P_t=200\times 5=1000\text{W}$
Hence, option (b) is the correct answer.