# Simple harmonic motion -> amplitude independent of mass?

1. Jul 10, 2009

### intelinside

simple harmonic motion --> amplitude independent of mass?

I know that acceleration is directly proportional to displacement, but opposite in sign, and that acceleration and displacement are related by the square of the frequency. But i was wondering if amplitude is independent of mass in simple harmonic motion? So for a spring hanging from the ceiling if i double the mass of the object hanging from the ceiling will the amplitude change? What about for a pendulum?

I was under the impression that if i increase the mass the amplitude will decrease and amplitude would increase when i take mass off, but then i hit this problem in ExamKrackers Physics 1001 #742 (MCAT Book) and it says that since the force on both sides of the simple harmonic motion are proportional to the mass, the acceleration, and thsu the distance aka amplitude, will remain the same when the mass is increased or decreased.

Mind you this simple harmonic motion is using friction on either side to keep it moving back and forth but I still would have thought it would have been decreased. According to the way they present their explanation I would conclude that in pendulums the amplitude would be independent of mass as well, since the restoring force is mgsin(theta). In springs, howver, the force is F=-kx so an increase in mass would decrease the amplitude..

Can someone please clarify this for me??

Last edited: Jul 10, 2009
2. Jul 10, 2009

### turin

Re: simple harmonic motion --> amplitude independent of mass?

One thing to keep in mind with a mass on a spring hanging from a ceiling is that, larger mass will relax to a lower eq position, making the eq length of the spring longer. However, assuming that the k of the spring is independent of the stretched length, the amplitude of any osciallation should depend only on the initial conditions of the oscillation. If you take the same spring and hang it from the ceiling with no mass, and then attach a mass and let it go from there, then the larger mass will give you larger oscillations. This is not an intrinsic property of the simple harmonic motion; it is simply due to the fact that the larger mass was initially displaced further from its eq position.

One thing to keep in mind about pendula is that the restoring force is actually nonlinear in the position, even for an ideal pendulum. The motion of the bob is only approximated by simple harmonic motion for small oscillations. However, again, since gravity is essentially the restoring force, the mass should factor out of the motion (ideally). A nonideal pendulum would, e.g., include a mass for the arm. Then, a larger bob mass would shift the center of mass closer to the ideal point (and have a decreasing effect on the frequency for a given amplitude).

3. Jul 10, 2009

### intelinside

Re: simple harmonic motion --> amplitude independent of mass?

According to the way they present their explanation I would conclude that in pendulums the amplitude would be independent of mass as well, since the restoring force is mgsin(theta), as well as this:

period = 2 pi sqrt(L/g) = 1/f

and acceleration and displacement are related by the square of the frequency
a = w^2 * x = (2 pi f)^2 * x;

therefore there is no correlation between mass and frequency thus no correlation with amplitude/distance in pendulum

In springs, however, the force is F=-kx so an increase in mass would increase the amplitude.

period = 2 pi sqrt(m/k) = 1/f

and acceleration and displacement are related by the square of the frequency

a = w^2 * x = (2 pi f)^2 * x; there is a correlation between mass and frequency here

so for the spring your saying that the amplitude would increase with increasing weight but for a pendulum the amplitude would remain the same, even if i originally have mass on there and then add an additional mass?

Last edited: Jul 10, 2009
4. Jul 10, 2009

### turin

Re: simple harmonic motion --> amplitude independent of mass?

I make no claim. I wanted to show you some realism, i.e. consider what would happen if you actually wanted to test this in the real world. Bottom line: the equations that are typically used in the simplified verions of both cases do not imply any intrinsic dependence of amplitude on mass.

5. Jul 11, 2009

### vin300

Re: simple harmonic motion --> amplitude independent of mass?

An important point here is that the period does not depend on its mass
frequency is not affected by mass if the length is large and angle is less.
In the spring you must note a=-k/m(x)
acceleration decrease with mass amplitude increases
In the pendulum a=-g/l(x)
does not depend on mass.

Last edited: Jul 11, 2009