Linear acceleration on a rotating body

[spaghetti3451]

What is the difference between tangential and radial acceleration for a point on a rotating body? As far as I know, the tangential acceleration changes the magnitude of the linear velocity of the point and the radial acceleration changes the direction of its linear velocity. But I don't understand why. And why do they have to point in the directions they do?

 

[Simon Bridge]

If you push a body with a force that acts directly through the center of mass, the body just accelerates in a line and does not turn. You can try this stuff out on a table-top.

If you push it off-axis though, it will rotate as well as move off in a line.
The line it moves off is the one through the point the force is applied and the center of mass. The component of the force perpendicular to this rotates it.

So far so good.

To get the object to be rotating but not translating, you have to apply the force tangentially. That makes sense - you end up pushing the edge in a circle so it is no surprise to find it rotating. If you keep pushing, the rotation gets faster and faster which is also no surprise - you are pushing in a circle. It is only at an instant of time that we say your force is tangential.

Stop pushing, however, and (if nothing else happens) the body keeps spinning. This is just the law of inertia in action. Thing is, the bits of the body don't want to go in a circle, they want to go in a line. The reason they don't is because the bits of the body are all stuck to the other bits of the body.

The force needed to hold the bits together has to be big enough to change the direction that the bits are trying to go in. That force has to point to the center - because that is the direction of the change in velocity of all the bits.

And a = F/m.