A particle with mass m on a straight line is being pushed and pulled by a force

Question

A particle with mass m on a straight line is being pushed and pulled by a force of magnitude F(t)=F0 sin(wt)

F0 is the amplitude of the force and w is the frequency at which it oscillates. m,F0, and w are constants.

a) what is the acceleration of the particle? If the initial velocity was v0 and if the particle starts at x=0 then where is the particle at time t?

b)What effect does an increase in the frequency w have on the velocity v(t) of the particle?

A particle with mass m on a straight line is being pushed and pulled by a force of magnitude F(t)=F0 sin(wt) F0 is the amplitude of the force and w is the frequency at which it oscillates. m,F0, and w are constants. what is the acceleration of the particle? If the initial velocity was v0 and if the particle starts at x=0 then where is the particle at time t? What effect does an increase in the frequency w have on the velocity v(t) of the particle?

Explanation / Answer

a).

a = F/m = (F0/m)*sin(wt)

Integrating both sides v = -(F0/(m*w))*cos(wt) + c

Since v at t =0 is v0

Hence, c = v0 + F0/(m*w)

So, v = v0 + F0/(m*w) - (F0/(m*w))*cos(wt)

Integrating again, x = ( v0 + F0/(m*w))*t - (F0/(m*w*w))*sin(wt)

Satisfying x = 0 at t = 0

b).

dv/dw = -F0/(m*w*w) + F0/(m*w*w)*cos(wt) + (F0/m)*sin(wt)

Which shows that dv/dw is greater than 0

Hence, v increases with increase in w..

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