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.

 

[paranormal]

I can provide some guidance on this question. First, it's important to note that simple harmonic motion (SHM) is a type of motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. This type of motion can be described by a sine or cosine function.

Now, to address your questions:

1. If the particle is given a velocity U towards the center of SHM when it reaches maximum displacement, the SHM will continue but with a different amplitude and phase. The equations used to describe SHM will still be the same, but you will need to consider the new initial conditions (X=a when t=0) in your calculations.

2. If you need to derive the equations again, the center point of SHM will still be the same. However, the amplitude and phase will be affected by the new initial conditions. The displacement and maximum displacement will also be different.

3. When integrating to get the equation of motion, it's important to consider the initial conditions. In this case, it seems you have taken the initial conditions as x=a, v=u and t=0. However, it's not clear what you are integrating with respect to. If you are integrating with respect to time, then your equation should be x=acos(ωt), not asin(ωt). It's also important to note that the equation of motion for SHM is a simplified form, and in reality, the motion may be more complex.

In conclusion, if the initial conditions of the SHM change, the equations used to describe it will also change. It's important to consider the initial conditions and the type of motion (sine or cosine) when deriving the equations. Additionally, it's important to be clear on what you are integrating with respect to when finding the equation of motion.