Q & A: Wobbling pivots and energy loss

There is a point mass rotating around a pivot. From what I understand, if the weight of the pivot is infinite, then in a frictionless environment the rotation of the point mass would never slow down. However, if the mass of the pivot was very light, then the pivot point would wobble as the centrifugal force of the point mass pulled it in different directions as it rotated. In this case, the point mass would slow down over time. So given the mass and angular velocity of the point mass and its radius from the pivot, and the mass of the pivot point, how quickly would the rotation of the point mass slow down in a frictionless environment?
- Michael Daly (age 23)
Dallas, TX, USA

Hi Michael,

That sounds like a very interesting practical problem! It sounds as if you have observed this behavior in a real situation, which is the very essence of what science is all about.

The easiest way to think about the moving object slowing down in this case is that it is somehow losing energy. A point mass rotating (maybe held with a string or a rod or something) around a pivot, neglecting air resistance and sliding friction of whatever holds the string or rod to the pivot, as you say, should not slow down, as long as the pivot is stationary. The pivot may be held fixed by nailing it down, that is, attaching it to something very heavy, or it may be heavy itself.

The case in which the pivot does move is more complicated. If the point mass and the pivot do not have any other forces on them, they will both rotate around a common center of mass, and the string will pass through this center of mass point. The center of mass is just the weighted average of the location of all the mass in the system (there need not be any actual mass at this location). The tension on the string will always point along the string, and the instantaneous motion of the point mass and the pivot will be perpendicular to the string.

In this case, no energy is lost, as all the forces involved are perpendicular at all instants to the velocities of the objects they push or pull on, and the forces do no work. Pushing a heavy object along frictionless ice requires very little energy, as gravity does not need to be overcome (some kinetic energy has to be imparted, but if you are patient, you can get away with as little as possible). This is because the motion of the object is perpendicular to the force of gravity and no work has to be done to overcome gravity. Lifting the object is different!

This situation appears in astronomy all the time. Planets orbit stars, and stars orbit stars, without the loss of energy even though the "mass" and the "pivot" both orbit a common center of mass (some tidal effects may cause energy loss, but these don’t exist for point masses).

Your situation is more complicated because it appears as if your pivot may be rubbing on something when it moves. This rubbing introduces energy loss due to friction. You can overcome this energy loss either by reducing the friction force, by oiling the surfaces that are rubbing, or by eliminating the motion by making the pivot heavy or by nailing it down.

Tom

(published on 10/22/2007)

In a real world situation, typically in mechanics, a loose bearing is a point of lost motion and in turn lost energy. It is important to "set" the bearing load when doing maintenance. Whether it be plain bearings or bushings or ball / taper bearing this is true. With a loose bearing or bushing, how can one determine the "sweet spot" without over tightening the bearing or it being too loose? Of course this is a clean and properly lubricated bearing set!
- Rod Wallberg (age 50+)
Ishpeming, Michigan, USA

Mike and I are not knowledgeable in this field so I asked a local expert, Prof. Norman Miller, for his advice. Attached are his comments.
LeeH

Except for extremely loose bearings, bearing adjustment is normally not an issue in terms of machine function or energy loss. Bearing adjustment does have a significant impact on bearing life. Usually bearing life is maximized for rolling element bearings like ball and roller bearings when they are lightly preloaded, however, bearing life is usually not significantly reduced for installations with zero clearance (zero preload) or even for slight clearance. In most modern machines, the bearing mountings are manufactured such that adjustment of the bearings is not possible. An exception to this rule is the case of tapered roller bearings (often used as automotive wheel bearings).  The issues are discussed on page A140 of the Timken Products Catalog (Engineering) available at http://www.timken.com/en-us/products/Pages/TimkenProductsCatalog.aspx. Unfortunately, the optimal setting is difficult to specify without careful testing. Frequently, the cold bearing setting, either a preload or endplay, will change significantly when the machine reaches operating temperature. This makes it impossible to specify a general rule of thumb. However, it is best to err on the side of excess endplay because excessive preload can cause drastically shortened bearing life and in the extreme can produce deformation of the bearing races (brinelling, see http://www.answers.com/topic/brinelling).

Norman M

(published on 11/15/11)