Question

A particle of mass $m$ moves under the influence of the force $\overrightarrow{F}=a(\sin \omega t\hat{i}+\cos \omega t\hat{j})$ where $a,\omega$ are constants and $t$ is time. The particle is initially at rest at the origin. Find the instantaneous power given to the particle.

Answer 1

$F=a(\sin \omega \hat{i}+\cos \omega \hat{j})$

Acceleration$\overrightarrow{a}=\frac{F}{m}=\frac{a}{m}\left(\sin \omega t.\hat{i}+\cos \omega t.\hat{j}\right)$

As we know, $\frac{dv}{dt}=a$

$dv=adt$

$\overrightarrow{v}=\int \overrightarrow{a}.dt=\frac{a}{m}\int \left(\sin \omega t.\hat{i}+\cos \omega t.\hat{j}\right)dt=\frac{a}{m}(\frac{-\cos \omega t}{\omega }\hat{i}+\frac{\sin \omega t}{\omega }\hat{j})+kt=0,\overrightarrow{v}=0k=\frac{a}{m\omega }=\overrightarrow{F}.\overrightarrow{v}=a\left(\sin \omega t.\hat{i}+\cos \omega t.\hat{j}\right).\frac{a}{m}(-\cos \omega t\hat{i}+\sin \omega t\hat{j}+\hat{i})=\frac{a^2}{\omega m}(-\sin \omega t\cos \omega t+\cos \omega t.\sin \omega t)+\frac{a^2}{\omega m}=\frac{a^2}{\omega m}$