# A particle of mass m moves under the force given by F = a(sinωt i +cosωt j). a,w are constants and t is time.It is intially at rest at the origin.What is work done on the particle up to time t and the instantaneous power given to the particle at time t?

May 23, 2018

$W = {a}^{2} / \left({\omega}^{2} m\right) \left(1 - \cos \omega t \setminus\right)$

$P = {a}^{2} / \left(\omega m\right) \sin \omega t \setminus$

#### Explanation:

By definition:

$\delta W = \boldsymbol{F} \cdot d \boldsymbol{r} q \quad = \boldsymbol{F} \cdot \frac{d \boldsymbol{r}}{\mathrm{dt}} \setminus \mathrm{dt} = \boldsymbol{F} \cdot \boldsymbol{v} \setminus \mathrm{dt}$

So it would be helpful to have the velocity vector .

Velocity

$\boldsymbol{F} = m \boldsymbol{a}$

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

$\boldsymbol{v} = \int \boldsymbol{a} \mathrm{dt} = \frac{a}{m} \left(- \frac{1}{\omega} \cos \omega t \boldsymbol{\hat{i}} + \frac{1}{\omega} \sin \omega t \boldsymbol{\hat{j}}\right) + \boldsymbol{C}$

$\boldsymbol{v} \left(0\right) = \boldsymbol{0} \implies \boldsymbol{C} = \frac{a}{m \omega} \boldsymbol{\hat{i}}$

implies bb v = a/(omega m) (1- cos omega t) bb hat i + sin omega t bb hat j)

Work

$W = {\int}_{0}^{t} a \left(\sin \omega \tau \boldsymbol{\hat{i}} + \cos \omega \tau \boldsymbol{\hat{j}}\right) \cdot \frac{a}{\omega m} \left(\left(1 - \cos \omega \tau\right) \boldsymbol{\hat{i}} + \sin \omega \tau \boldsymbol{\hat{j}}\right) \setminus d \tau$

$= {a}^{2} / \left(\omega m\right) {\int}_{0}^{t} \setminus \sin \omega \tau - \sin \omega \tau \cos \omega \tau + \sin \omega \tau \cos \omega \tau \setminus \setminus d \tau$

$= {a}^{2} / \left(\omega m\right) {\left(- \frac{1}{\omega} \cos \omega \tau \setminus\right)}_{0}^{t}$

$= {a}^{2} / \left({\omega}^{2} m\right) \left(1 - \cos \omega t \setminus\right)$

Power

$P = \frac{\mathrm{dW}}{\mathrm{dt}} = {a}^{2} / \left({\omega}^{2} m\right) \left(0 + \omega \sin \omega t \setminus\right)$

$= {a}^{2} / \left(\omega m\right) \sin \omega t \setminus$