A New Look Towards the Principles of Motion - Section 5, Forces involved in Rotational Motion

[Arindam Banerjee]

A New Look Towards the Principles of Motion

A book by Arindam Banerjee for fresh and keen young minds,

Section Five

The Forces involved in Rotational Motion

So far we have dealt with momentum, forces, etc. in the one-dimensional world. While convenient for conceptualization and ease of explanation, the world is really three dimensional. The understanding we have, from the earlier analysis, must be projected into the three-dimensional work. The true work of the keen engineer, then begins! In this book we will not be going into the advanced mathematics that naturally follow. We will, however, interest ourselves just a little bit into the subtleties of rotational motion, which has to be two-dimensional at the least.

Now, take the motion of the earth, revolving around the sun. The *only* force involved is gravity. But one may say, if gravity is the only force to be considered, then why does not the earth fall into the sun?

The answer is very simple. The earth is always "falling into the sun" but still maintains more or less the same distance from the sun because it has a certain velocity which is tangentially directed away from the line joining the sun and the earth at any given time. So the gravity pushes the earth at any time towards the sun in a straight line, but over that time the earth because of its tangential velocity also moves a certain distance perpendicular to that line. So over a small amount of time, the earth moves inwards and also moves tangentially; and the resulting fact is that after that small amount of time the earth is still the same distance away from the sun, though it has of course moved away from the original position. Of course, a lot depends upon the velocity and distance from the sun, and interaction from other bodies, and so on.



With this sort of analysis, we need not bother about the so-called centrifugal or centripetal forces, that are referred to in physics textbooks as “pseudo forces”. As we know, with current understanding, the centripetal force here is gravity, while the centrifugal force is what keeps the centripetal force in check, balances it that it. This centrifugal force, apparently, has to arise from Newton’s Third Law – it is referred to in physics texts as a “pseudo force”.

Take some further examples of the conceptual (but not really bogus) centripetal and centrifugal forces involved in rotational aspects that are commonly found. Such as the motion of a coin on a rotating gramophone disk (a rarity these days, but once they existed in profusion), and the motion of any objected rotated (say by hand) by a string.

Centrifugal force seems to be self-evident in the former case. You put the coin on the disk, start the motor, and lo the coin shoots outwards! Surely there was a force that pushed it out because it was now moving, and that force is centrifugal force! The very term "centrifugal tendencies" arise from the evidence of such-like phenomenon in nature.

But really, there is no centrifugal force, only the appearance of same. The only force involved is the weak force (or inter-atomic force) manifesting in this case as friction between the disk and the coin. Let us now consider the subtleties involved in this.

If there is no friction, the coin will be static - the disk will simply move under it, and the coin will stay put. If there is too much friction, then the coin will move with the disk, like it were stuck with Velcro or honey. It is only when there is some but not that much friction, that the coin will move outwards. Then, the friction will make the coin move tangentially. Because it is moving tangentially over small intervals of time - this coin now is having a greater radius than before, with the passing of incremental time, and angular motion. Velocity addition effects are taking place (as explained in my formula linking mass and energy) and ultimately the coin simply shoots off the disk.

Similarly, for the second example. If a stone or ball is tied to a string, and the string is given a rotational motion by the hand, then the only force is the tension within the string (again a weak force, relating to the atoms comprising the string). Here again, the ball/ stone is "always flying away tangentially" but also "getting pulled back by the force in the string, this force being always at right angles to the tangential motion of the ball" so that a constant radius is getting maintained. As the ball rotates faster, the tension in the string increases.

Well, these are new ways to look at rotational motion, and they are based upon the fundamental forces accepted by the honest physicists. One can very well do without the confusing notions of centripetal and centrifugal forces.

I can now mathematically describe the motion of a coin
on a rotating turntable as,

r * secant(del-theta) = r + del-r1,

(r + del-r1) * secant(del-theta) = r + del-r1 + del-r2,

and so on,

till r + del-r1 + del-r2 + ... + del-rn = R

where r is the initial radius where coin is placed on the disk which is moving at angular speed omega radians per second,
and R is the radius of the disk and del-theta is the angle that is continuously getting displaced with very small time interval del-t, so that

del-theta = omega * del-t * friction_factor

and the del-r1, del-r2, etc are the incremental radial increases with del-t and the friction-factor varies between 0 and 1 and is gamma(t)/omega, where gamma(t) is the angular speed of the coin as a function of time, and its actual values may be found from experiment. It may be assumed to be a constant, for ease of explanation. Note that if gamma is zero, it means that there is no friction, and thus the del-r terms do not happen, as the disk is simply sliding under the coin. We do not need centrifugal force or centripetal force, just the friction coefficient between 0 and 1 to describe the motion of a coin on a rotating gramophone record.

Now what happens if the coin is sort of stuck to the disk? The situation is similar to what we may experience in a “centrifuge” in an amusement part. There, we are expected to sit tight against the recessed circular inner wall, which rotates fast, and against which we get pressed tight. We may even get lifted off our feet. This is the most direct evidence of centrifugal force that we experience!

The sticky thing that fixes the coin to the disk, is equivalent to the rotating wall. Here the del-R increments would have happened, if the coin had not been stuck. But, as the coin is stuck, not a single del-R happens – the sticky stuff comes under strain instead. The coin would have travelled del-r in time del-t, so the linear acceleration can be found out. From this value of acceleration, the force, taking the mass of the coin into account, can be calculated.

We should have stuff on us that stick well when we take a ride in the centrifuge. Otherwise, the wall will slip under us, and the next guy to us will hit us! If we do have sticky clothes, then the centrifuge wall that is rotating, will rotate us along with it. But the centrifuge is not a travelator, so it is curving. The curve is always arresting the linear, tangential motion at every instant – and it is this curve which is giving us the impression of a force acting upon us. So for the same angular velocity, if the radius is great, we do not feel such a force. If a travelator is seen as a centrifuge of inifinte radius, we thus feel no “centrifugal” force at all. As the radius becomes small, we feel the “centrifugal” force more and more – while actually it is the steepness of the curve’s angle, caused by the smaller radius, that is doing the trick.