# Uniform Circular Motion w/ Tension and Friction

So I had a problem today which I couldn't make any sense of. I don't have access to it at the moment but this is a pretty accurate approximation.

Basically, a mass (m) is swinging horizontally on a surface in uniform circular motion. It is attached to a central point by a string of given length (r). The string is exerting a given tension force (T). The period is essentially given as it can be derived from other given values. The question is, what is the force of friction the mass experiences.

So we had some lively debate on the matter amongst my classmates and couldn't come to a satisfactory consensus (especially since none of us are particularly good or knowledgeable at physics). It basically boiled down to the following:

• Friction is tangential to the circle and opposite the direction of the velocity; the tension force is at an angle as a result (to balance out friction with a component)
• Friction is in the same direction as tension (ie. radial, inwards toward the centre) and supplements it as a centripetal force

We also couldn't figure out if the friction in question is static or kinetic. By comparison with a car making a turn, it would be static friction, whereas it was also argued that the mass was sliding, in which case it would have been kinetic (? I'm really unsure about the arguments here).

I'm not altogether too sure of the validity of anything above, as we sort of just speculated based on one or two examples we did in class that weren't too clear themselves. If someone could please explain how to solve this problem (in preferably simple terms because high school), it'd be much appreciated.

• Friction is always normal (i.e. perpendicular) to the contact surface and opposite the velocity vector. – Gert Nov 5 '16 at 2:17
• @Gert Wait what? Isn't friction parallel to the surfaces in contact? – M. Enns Nov 5 '16 at 2:21
• Oops. Yes. $F_F$ perpendicular to the Normal force, of course. My bad. – Gert Nov 5 '16 at 2:38

Friction always opposes relative motion (or the tendency towards relative motion) between the two surfaces in contact.

If there is relative motion the friction is kinetic. If there is no relative motion but forces are tending to cause relative motion, the friction is static.

• Could you please explain what you mean by "relative motion"? As in, relative to one, the other is moving? – YogrtMan Nov 5 '16 at 2:16
• Yes, that is correct. – sammy gerbil Nov 5 '16 at 2:22
• So in this case, since the mass is not moving at the same velocity as the table, it is kinetic friction, yes? Thank you very much for your help. – YogrtMan Nov 5 '16 at 2:32
• Yes, if there is relative motion between the mass and the table, and they are in contact, then there is friction and it is kinetic. (The details of the motion are not clear in your question, as M Enns points out.) – sammy gerbil Nov 5 '16 at 2:44

It sounds to me, the way you describe the problem, that the friction would have to be zero.

If the motion is described as "uniform circular motion" then the object is moving with a constant speed, this tells us that forces in the direction of motion and opposite to the direction of motion must be balanced. Friction would act to oppose the motion so, to maintain a constant speed, there would have to be some forward force. But if the string is attached to the centre of rotation it can only supply a radial force i.e. perpendicular to the motion. If the tension force is strictly towards the centre it can't balance the force of friction so either there's zero friction or the object slows down.

Now, if you were going to try to actually set up this situation by tying a string to something and making it go around a circle on the floor with a constant speed you would have to move your hand as you hold the string so that you lead the object a bit. In this case it would be like the option you describe in your first bullet point where there would be a tangential component to the normal force that would balance out the friction.