Solving Simple Harmonic Motion Question

[acmmanoj]

I am having a question and tries to solve a problem for days. Consider general SHM. When the particle reaches to the maximum displacement, a if a velocity U is given to the particle towards to the center of SHM, (keeping the same force mω^2x)

1. What would happen to the SHM...is it same or can i use same equations or should i derive equation again

Should i consider this as new SHM ( X=a when t=0) or should i continue from same SHM (X=0 when T=0)

2. if i derive again, which point should i considered as center, what will happen to the displacement and maximum displacement...is it same or difference

3. when tries to get equation of motion as x=asin(ωt) from intergartion it produces a very complex equation. (in intergration i took, when x=a , v=u and x=a t=0)

 

[Simon Bridge]

You are saying that when the pendulum reaches it's maximum displacement you give it a kick towards the center?

1. This ads energy to the system, resonantly.
This is no longer SHM - it is now a driven harmonic oscillator.

2. Without damping you are solving something like:

$$m\frac{d^2x}{dt^2}+kx=f(t)$$
... where f(t) is the applied force.

You can simulate a series of kicks by a sequence of Dirac-delta functions for the specific impulse. The center is still the same, the mass will come back further.

3. ... and yes, the equation can get quite complicated.

 

[acmmanoj]

Thanx.. this is helpful

Can i use conservation of energy to solve the problem?

 

[Simon Bridge]

Kinda - the oscillator is no longer a closed system though.
Each kick then provides a bit extra KE, so whatever provides the restoring force has to store more as PE, and so the amplitude of the motion increases. The characteristic frequency of the motion won't, except perhaps at high amplitudes, since you are timing the kicks to it. At high amplitudes, pendulums are no longer simple, and springs may exceed their elastic limit, for eg.

Anyhoo - that would give you a sequence of sine-portions with a sharp change at each kick.