Simple Harmonic Motion and acceleration

In summary, the equation for maximum acceleration in simple harmonic motion shows that the acceleration is proportional to the amplitude and the square of the frequency, but there is no fixed relationship between the three variables. They can vary independently and can change depending on the system in question.
  • #1
Jimmy87
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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!
 
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  • #2
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.
 
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  • #3
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
 
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  • #4
Einj said:
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?
 
  • #5
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
 
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  • #6
kq6up said:
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?
 
  • #7
Jimmy87 said:
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.
 
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1. What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction of the displacement. This results in a repeating back-and-forth motion around the equilibrium point.

2. What are the factors that affect the acceleration of an object in SHM?

The acceleration of an object in SHM is affected by three factors: the amplitude (maximum displacement from equilibrium), the frequency (number of cycles per unit time), and the mass of the object. As the amplitude and frequency increase, the acceleration also increases. However, a higher mass will result in a lower acceleration.

3. How is acceleration calculated in SHM?

The formula for calculating acceleration in SHM is a = -ω^2x, where ω is the angular frequency (2πf) and x is the displacement from equilibrium. This formula shows that acceleration is directly proportional to the displacement and inversely proportional to the angular frequency squared.

4. What is the relationship between acceleration and velocity in SHM?

In SHM, acceleration and velocity have a sinusoidal relationship. This means that as the acceleration reaches its maximum value (either positive or negative), the velocity will be zero. Similarly, as the velocity reaches its maximum value, the acceleration will be zero.

5. How is SHM related to simple pendulums and springs?

Simple pendulums and springs are commonly used examples of SHM. In a simple pendulum, the restoring force is provided by gravity, while in a spring, the restoring force is provided by the elastic properties of the material. Both of these systems exhibit SHM when disturbed from their equilibrium position.

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