Rotational mechanics theory

This is a statement given in my text book:

"The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axis need not be stationary."

I don't understand this properly. Please explain.

The thing is actually quite simple.

A rigid body, by definition, is such that preserves its shape along the movement; it is not deformable. ...and that means that the distances between all its constituent particles remains fixed.

Consequently, the motion can only be such that preserves that shape. If the motion deformed the body, it would be okay, but it couldn't be labeled as "rigid body".

So the movement will conserve the shape of the body, and there are only two possibilities for this:

1. Translation (A global) translation of all particles at the same time)
2. A rotation around an axis, (all particles rotating at the same time throug an axis).

These are the two only possible movements that preserve the shape. A combination of both would also do so.

And finally, the text is saying that (1) is equivalent to the movement of the CM. A global translation can be described as the translation of any point. That's because "rigid body" implies that the rest of particles will move in the same way, for the shape to be maintained.

So you can choose any point to say "it moves like this". If you can choose any point, the most intelligent chose is picking the CM. It's like saying "the car is where its CM is", altough you could say "the car is where the tip of the front bumper is".

• Re "the car is where the tip of the front bumper is" -- That's oftentimes exactly what is done to describe the motion of a robot or a spacecraft. Describing the motion of the center of mass is useful for introductory physics problems, not so useful for describe the motion of a robot or spacecraft. Robots have joints that constrain the motion of the robot, and spacecraft toss off 90% of their mass during launch. Apr 21, 2018 at 21:02
• What's bugging me is that an instantaneous axis of rotation need not be passing through the center of mass always. But its given here that it always passes through the com. Am I missing something ? Apr 22, 2018 at 1:12
• Indeed, the body can rotate about any axis. The thing is that this is equivalent to a sum of (1) and (2). For example, you can say that teh earth orbits around the Sun, or you can say that it's a sum of "displacement" + "rotation around its own center until it faces the sun again". Apr 22, 2018 at 10:42

This statement seems to be talking about several things. First, the author seems to be mentioning the fact that a barbell spinning about some kind of axis might very well be traveling through space (these axes need not be stationary). Also, they seem to be hinting at the parallel axis theorem.

Lastly, I agree that the author is using rather stilted words to describe a rather simple topic.

If it does not interact with other objects, a rigid body can be reduced to its center of mass. Then you can describe the motion of this point as the motion around an axis (in case of straight travel it's an infinitely far away axis). If for some reason you need to know more about what the constituent masses of the rigid body are doing, you can describe the motion of the object as a dance around an axis that goes through the center of mass. This axis can behave rather weird, for example if you throw a brick, that bricks' center of mass will go in a parabola, but the brick will probably tumble, because it has three very different main axis, and rotation is only stable around the longest and the shortest. Spin around the middle axis will soon become spin around one of the others. --- I said the center of gravity will move in a parabola - this is only describable as a rotation if the axis of the rotation moves nearer to the object all the time. So both axis move, and we can seperately describe the rotations around those axis.