# Centripetal Force on a ball kept on a rotating disc

Suppose I keep a ball some distance $$r$$ away from the centre of a rotating disc. Let's assume I keep the ball gently and slowly so as to not impart any velocity to the ball.

If the disc were to be frictionless, we know the ball would stay stationary in the ground frame and would move opposite to the direction of rotation of the disc in the disc's frame. I'm aware that the motion of the ball in the disc's frame can be explained using the Coriolis Force and Centrifugal Force in the rotating frame.

Now, suppose the disc is rough and is rotating with some (not too large) angular velocity. Now, we observe the ball rotates with the disc in the ground frame and is stationary in the rotating frame. My question is, what exactly causes the ball to exhibit circular motion in this case? I'm pretty sure it's friction but why will friction acts radially inwards if the relative velocity of the ball is tangential to the circular disc? Is this friction static or kinetic? Also, if I used some kind of cube or 'non-rollable' shaped body, what changes would occur?

Consider an object sitting on the rotating disc. If the friction is suddenly lost, the object will move radially outwards. This is why the friction has to be pointing radially inwards.

In more detail, yes, when you draw the instantaneous velocity vector, that would be tangential to the instantaneous position. However, if you draw it out as a straight line, which is what it would do in the absence of forces due to Newton's 1st Law, then you see that the tangential motion would lead to an increase in radial coördinate value.

That is, the rate of change of velocity is not tangential, but rather radial. i.e. Acceleration is radially inwards. By Newton's 2nd Law, we thus need to supply radially inward force in order for things to stay in a circular orbit. Hence why friction is pointing radially inward.

Relative to the rotating disc, the object is always tending to move radially outwards, and so friction will be pushing radially inward.

• I understand acceleration is radial, but isn't the direction of friction of velocity always opposite to the direction of velocity? Commented May 23, 2023 at 7:34
• This problem made it clear it is not so. You can see it in two ways. Either you consider the relative velocity of the object and the point on the rotating disc right beneath it, and that lets you see that it is actually stationary relative to the disc, and thus the friction can be pointing any direction it wants---of course, it would be opposing any attempted relative motion, hence radially inwards. Or you can consider rotating frame. Commented May 23, 2023 at 7:59
• This made it a little more clear... But can you please explain how the relative motion is radially outwards? Commented May 23, 2023 at 8:05
• If possible, can you please provide a more general explanation Commented May 23, 2023 at 8:05
• Okay, I understand it now... thanks! Commented May 23, 2023 at 8:33

The reason is that the direction of the velocity changes but its magnitude does not.

Thus no force is "required" to change the magnitude but a force is required to change its direction.
That change of direction (centripetal acceleration) is towards the centre of rotation and it is produced by a friction force acting towards the centre of rotation.

. .; . . . if I used some kind of cube or 'non-rollable' shaped body, what changes would occur?

None really.

• But how can friction act inwards if motion is tangential? Commented May 23, 2023 at 7:13
• The motion might be tangential but the direction of the tangent changes. Commented May 23, 2023 at 7:29