Simple Harmonic Motion and acceleration

[Jimmy87]

Hi, I have a few questions relating to the equation for maximum acceleration for SHM:

amax = A (2 x pi x f)^2 where amax = max. acceleration, A = amplitude, f = frequency.

How are these variables supposed to be interpreted when you relate them to each other. For example, is A inversely proportional to the frequency squared as the equation implies? So, if you doubled the frequency, would the amplitude go down by a factor of 4? This reasoning would involve holding amax constant which itself depends on A so I'm not sure whether what I said is justified. What happens to the variables in the equation if you do increase the frequency?

Thanks for any help given!

 

[Einj]

The frequency, amplitude and phase of an harmonic oscillator are independent quantities. They can vary from system to system and if the oscillator is free (no damping or driving force) they are fixed for ever. There is no reason why in you equation if, for example, you double the frequency the maximum acceleration should remain the same.

 

[kq6up]

Acceleration is proportional to A, and it is proportional to the square of the frequency, so if the frequency is doubled the maximum acceleration is quadrupled.

Chris

 

[Jimmy87]

The frequency, amplitude and phase of an harmonic oscillator are independent quantities. They can vary from system to system and if the oscillator is free (no damping or driving force) they are fixed for ever. There is no reason why in you equation if, for example, you double the frequency the maximum acceleration should remain the same.

Thanks for the replies. Take, for example, a ruler suspended between two supports. You hang a mass in the middle and set it oscillating. You can increase the frequency by moving the two supports closer (essentially making the ruler shorter). Say that you move the supports closer such that the frequency has doubled what would happen to amax and A in this situation according to the equation?

 

[kq6up]

That is a different set of equations. The maximum spring potential energy is conserved even when you change the ruler. $PE_{max}=\frac{1}{2}kx^2_{max}$ where k is the spring constant for Hooke's law. When you move your supports, you stiffin the spring and k increases. That means $x_{max}$ has to decrease for energy to be conserved.

Chris

 

[Jimmy87]

That is a different set of equations. The maximum spring potential energy is conserved even when you change the ruler. $PE_{max}=\frac{1}{2}kx^2_{max}$ where k is the spring constant for Hooke's law. When you move your supports, you stiffin the spring and k increases. That means $x_{max}$ has to decrease for energy to be conserved.

Chris

That's interesting, thank you. Can you still apply simple harmonic motion equations to this? Surely it is still undergoing simple harmonic motion is it not?

 

[Redbelly98]

Thanks for the replies. Take, for example, a ruler suspended between two supports. You hang a mass in the middle and set it oscillating. You can increase the frequency by moving the two supports closer (essentially making the ruler shorter). Say that you move the supports closer such that the frequency has doubled what would happen to amax and A in this situation according to the equation?

According to the equation, the ratio amax/A would quadruple if the frequency is doubled. But more information is required before we could say whether amax quadrupled, or A was reduced by a factor of 4, or amax doubled while A was halved, etc.