Ball against a wall, statics.

[philnow]

Homework Statement

Here is a horrible diagram representing the problem:

The problem is to find the minimum coefficient of static friction between the ball and the wall so that the ball remains motionless.

Homework Equations

torque = r*F

The Attempt at a Solution

I've divided the tension force into x and y components, Tsinθ and Tcosθ respectively. Therefore the normal force (the wall pushing against the ball) is N = Tsinθ. The friction force is = uN = u(Tsinθ).

So now, because the ball is motionless, the two torques must cancel each other out. So Torque from the tension T(t) = (radius)*T and torque from the friction force T(f) = radius*Friction force = radius* uTsinθ. This gives u = (1/sinθ)... but I'm really not sure... also, how do I minimize this u?

Thanks in advance for any help.

 

[Doc Al]

Looks good to me. Setting the static friction force to equal to its maximum value μN (as you did) will give you the smallest μ. (Generally static friction ≤ μN.)

 

[tomcue]

I would first like to clarify that the problem statement is incomplete as it does not mention the mass of the ball or the angle of the wall with respect to the horizontal. These variables can affect the solution and should be provided for a complete understanding of the problem.

That being said, I will assume that the mass of the ball is negligible and the angle of the wall is 90 degrees with respect to the horizontal.

In this case, the minimum coefficient of static friction can be found by considering the forces acting on the ball. The ball is in equilibrium, so the sum of the forces in the x and y directions must be equal to zero.

In the x direction, we have the tension force Tsinθ and the friction force uN. Since the ball is not moving, the tension force must be equal to the friction force, so we can write:

Tsinθ = uN

In the y direction, we have the normal force N and the weight of the ball mg. Since the ball is not moving in the y direction, the normal force must be equal to the weight of the ball, so we can write:

N = mg

Substituting N = Tsinθ from the first equation into the second equation, we get:

Tsinθ = u(mg)

Solving for u, we get:

u = Tsinθ/mg

To minimize u, we can see that the value of sinθ is fixed, so the only way to minimize u is to minimize T or increase the weight of the ball. Therefore, the minimum coefficient of static friction is achieved when the tension force is minimized or the weight of the ball is increased.

In conclusion, as a scientist, I would suggest considering all the variables involved in the problem and providing a complete problem statement before attempting to find a solution. Additionally, it is important to clearly define the coordinate system and all the forces acting on the object in order to accurately solve the problem.