# Object with friction in circular motion caused by a string

A physics problem in my textbook reads:

A 0.40kg ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3m on a frictionless horizontal surface. If the cord will break when the tension in it exceeds 60N, what is the maximum speed the ball can have? How would your answer be affected if there were friction?

Obviously the first question is easy to calculate. But the second one gave me some trouble. The book answer states that friction would not affect the problem, however I believe it would. In order for a limp cord to accelerate the ball in uniform circular motion, the force moving the cord would have to go in a circle of its own if there were friction. Below I drew a picture of my idea of the problem. (The text in the middle says center of rotation). You can see that the net force has to be greater than it would otherwise have been because of friction. Assuming that the book answer is wrong though, I have another question about the diagram I drew. Would the force of friction act tangent to the circle as I have indicated below? Or would it behave differently?

• Yes, the direction of force friction is tangent to the circle, because it acts in the direction opposite to that of the motion, and the direction of motion is always tangent to its path. But your diagram is still wrong in how it depicts the tension force. – David H Sep 11 '13 at 19:51
• I agree, the tension does not counter act the friction. – ja72 Sep 11 '13 at 21:00

Instructors in introductory courses like to say "Always draw a picture first!" That advice is particularly valuable here because your drawing makes it clear exactly what you think is going on. That helps to determine where your thinking goes awry.

The key to this problem is that the tension can act only in the direction along the string, that is, radially. What you have labeled as "Component of tension that counteracts friction" must be zero. So the tension remains equal to the centripetal force, regardless of the friction. But if the "component of tension that counteracts friction" is zero, what counteracts friction? Nothing. The object will slow down.

Your confusion might be due to a misinterpretation of the problem. I think you are reading into it a requirement that the object be kept at the same speed. That would require a tangential component of the tension. You could get that by grabbing the string, and swinging it in a circle, as you illustrate in your drawing. Under that interpretation of the problem, your drawing and conclusion are correct. But based on the answer given, the author of the problem probably means for you to assume that the end of the string not attached to the object is fixed at the center of the object's orbit.

You understand the case with no friction. Let's look at the case with friction.

There are two forces acting on the ball: tension and friciton.

The friction comes from the interaction between the string and the talbe, and the direction of friction is always opposing the direction of relative motion of the two objects in contact. So the friction force points along the tangent to the circle. The strength of the friction force is proportional to the weight of the ball which is constant in this problem.

The tension force arises from the interaction of the ball with the string. It is important to note that the tension force must always point along the string, so that here the force points to the center of the circle. The magnitude of the tension force must be the centripetal force for the circular motion: $mv^2/r$.

Notice two things. One, that the tension force can't possible slow down the ball since it is directed perpendicular to the ball's direction of motion. Two, that is ok since the table is the one that is responsible for slowing down the ball through friction.

Now we have our answer. Since the force from the string is the same whether or not there is friction, the maximum speeds for the two cases will not be different.