What is the effect of the mass of an object undergoing circular motion, on the period of the rotation?

Question

Is the mass directly proportional to the square of the period? I mean, it makes sense logically that the mass of the object would affect how fast it spins. Can anyone provide a source?

Answer 1

You need to specify the physical circumstances better.

But WHY should you think that "it makes sense logically that the mass of the object would affect how fast it spins" ?! I DON'T make that kind of connection AT ALL!

[Thinking DIMENSIONALLY, mass has dimension [M], while period has dimension [T], and ' g ' for example, has dimensions [L] / [T]^2. So, given these physical facts, a relationship between ' g,' period and some LENGTH may be anticipated; but any attempt to involve any mass in it would also require using either that same mass again, or another mass in the problem, to get rid of it from the dimensional relationship.]

For example, if this object is undergoing circular motion under gravity on the end of a string, the MASS has absolutely NO effect on the period. (The gravitational force is proprtional to the mass, but since the mass factors out in determining accelerations, the kinematical considerations are then INDEPENDENT of the mass.)

The relevant variables COULD be: the length of the string, the angle it makes with the vertical, the speed with which it's undergoing its circular motion, etc., --- any or all of these could be useful givens in any particular problem, but definitely NOT the mass.

(Recall, for example, Galileo's classical observation that the period of a pendulum depends ONLY on the length of the string, and not on either the mass or the constitution --- lead, wood, steel, etc. --- or on the amplitude of the swing, provided that it's small.)

But perhaps that is NOT what you're envisaging. Whatever it is, it's a bit confusing because you use the following terms (i) "undergoing circular motion," (ii) "rotation," and (iii) "how fast it spins," which do not all describe the same dynamical or kinematical phenomena.

(ii) and (iii) are synonymous, but (i) is a quite distinct dynamical motion. This reminds me of students who confuse the two SEPARATE, QUITE DISTINCT motions of the Earth: (i) it REVOLVES around the Sun, but (ii) it ROTATES about its own axis.

So, please clarify what you're thinking about.

Live long and prosper.

Answer 2

I think that if you could a ten pound mass at the end of a rod fast enough it should lift you into the air. Are you dabbling in the perpetuo moto? If so, enjoy. I have.