# Explain the relative motion between two particles of a rotating rigid body

A rigid body is defined as a collection of particles in which distance between each pair of particles remains constant.

I was taught that the motion of any one particle of the rigid body with respect to any other particle of the same rigid body is circular. If distance between two particles is constant and both of them are on a rotating rigid body why would the motion of one particles be circular with respect to other,it should be at rest right?

The question I want to ask ,in more detail; now, is:

Let's say that earth is revolving around sun ,without any rotation about its own axis and if we look at earth from sun's frame of reference,earth would appear to be revolving around us with one face of it always towards us.

Now, if we look at sun from earth's frame of reference,how would it look?

• Would sun appear to be revolving around us? if it is so, then it is wrong right ? because if sun is revolving around earth then from earth's frame the back side of earth also gets sunlight ,which should not happen.

• If sun doesn't revolve around us then what would happen?

Consider a rigid body now

Let's say that the below rigid body rotates about its geometrical centre about an axis perpendicular to the paper

• consider two points A and B(some random points) on the rigid body and now join the points A and B

• As the body rotates the line AB also rotates about the same axis

Now,how would the motion of B look from A? and how would the motion of geometric center look from A?

• let's suppose that A is earth and B is sun,if you say that B follows circular path about A then you mean that sun follows circular path about earth then the sunlight should fall on earth's back also,but that shouldn't happen right? so how does this happen?

PLEASE TRY TO EXPLAIN THE PROBLEM USING THE SUN AND EARTH METHOD

• See here for a solution with two (and more) objects in orbit maintaining the same distance. Isn't the observation: Angular velocity of all points on rigid body is same, since all are covering same angular displacement in same time.
– Rob
Commented Jul 2, 2018 at 2:00

The key phrase is "with respect to". That means a frame of reference where one of the particles is fixed in space. Since the distance to the other is fixed, any relative motion of the other particle must be circular.

The quick answer is that Chasle's theorem requires the general motion of a rigid body to be a rotation about an arbitrary axis, as well as a parallel translation along this axis. This is a requirement of the fact that all particles must remain fixed relative to each other.

The fact that you are trying to explain this with the sun-earth system is has two pitfalls:

1. A particle isn't like a planet, because it has no notion of rotation or orientation for a particle.

2. Even if planets were particles, two particles do not make a rigid body. Two particles have more allowed motions than a rigid body has. You need three or more particles glued together to have a rigid body type of system.

First of all u are saying that one face is always towards sum , automattically means earth is rotating abouts it's own axis with same angular velocity as it is revolving around the sun.For the feel let's say a car is moving straight with its frontier part pointing north . If it's frontier part if do no point in same direction at any instant it means it must have rotated. Now initially say face pointing sun is pointing to north. After half revolution this face is pointing south. So earth must have rotated by π radian also.

Simply in rigid body , distance between 2 points do no change. Let's say body is only translating then any point of the rigid moves with same velocity so they all are at rest with respect to each other. Now let's a disc is purely rotating about its axis, then point at centre is at rest but other points are not and they will move in circles about central point with same angular velocity. So particles do circulation motion with respect to each other only if body is rotating. The angular velocity of any point with respect to any point is same.

Let's say earth and sun both rotates in circle , then they will rotate about their centre of mass with same angular velocity.(position of centre of cannot change as distance between planets do not change). Mass of sun is so high that it will very near to centre of mass. It can approximately be assumed that sun is at rest as it is almost lie on the axis. Any point on the axis is at rest.