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?

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Answer 1 · 2018-05-23T10:02:47

#W =a^2/(omega^2 m) ( 1 - cos omega t\ )#

# P =a^2/(omega m) sin omega t \ #

By definition:

#delta W = bb F * d bb r qquad = bb F * (d bb r)/(dt) \ dt = bb F * bb v \ dt#

So it would be helpful to have the velocity vector .

Velocity

#bb F = m bba#

#bb a = a/m(sin omega t bb hat i +cos omega t bb hat j)#

#bb v = int bb a dt = a/m (-1/omega cos omega t bb hat i + 1/omega sin omega t bb hat j) + bb C#

#bb v(0) = bb 0 implies bbC = a/(m omega) bb hat i#

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

#W = int_0^t a(sin omega tau bb hat i +cos omega tau bb hat j)* a/(omega m) ((1- cos omega tau) bb hat i + sin omega tau bb hat j) \ d tau#

#=a^2/(omega m) int_0^t \ sin omega tau - sin omega tau cos omega tau + sin omega tau cos omega tau \ \ d tau #

#=a^2/(omega m) ( - 1/omega cos omega tau\ )_0^t#

#=a^2/(omega^2 m) ( 1 - cos omega t\ )#

#P = (dW)/(dt) =a^2/(omega^2 m) ( 0 + omega sin omega t\ )#

# =a^2/(omega m) sin omega t \ #