The position of a mass(m), is moving attached to a spring with constant(k) in a

Question

The position of a mass(m), is moving attached to a spring with constant(k) in a friction-less medium. The mechanical vibration is caused by a periodic external force of frequency(w) and amplitude(F0).

a) For the periodic external force, the modeling choice is either sin or cos, which one of these two periodic functions would you choose, and why?

b) Assuming that the natural frequency of the motion is different from (w), find the position of the mass at any time(t). you can also assume that initially, the mass was at rest and that the system was at equilibrium.

Explanation / Answer

a). both is okay. But we tend to use cos so I ll use cos. b)we have that F=F0*cos(wt+phi). because the system was at equilibrium so that phi=-pi/2. F-kx=ma. so that F=kx+mx''. assume that x=A*e^(iwt+phi). so that x''=-A*w^2*e(iwt+phi) so that F0*cos(wt+phi)=A*(ke^(iwt+phi)-w^2*m*e^(iwt+phi)) so F0*cos(wt-pi/2)=A*(k-w^2*m)*cos(wt+phi). so that phi = pi/2 too. so F0=A*(k-w^2*m). so A=F0/(k-w^2*m). so x=(F0/(k-w^2*m))*cos(wt-pi/2)

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