A solid ball is released from rest and slides down a ramp that makes an angle of 50 degrees with respect to the horizontal. a) What is the minimum coefficient of static friction between the ramp and the ball for no slipping to occur? b) Would this coefficient of static friction be sufficient to stop a hollow ball (like a basketball or soccer ball) from slipping? Why or why not? c) Why does part a ask about the coefficient of static, not kinetic, friction?
To answer each part of your question:
a) To calculate the minimum coefficient of static friction between the ramp and the solid ball for no slipping to occur, we need to consider the forces acting on the ball. There are two main forces at play: the force of gravity pulling the ball down the ramp and the force of static friction opposing the sliding motion. The frictional force acting on the ball can be expressed as the coefficient of static friction (μs) multiplied by the normal force (N) acting perpendicular to the ramp.
The normal force can be found by considering the force components acting on the ball. We have the force of gravity acting vertically downward, which can be decomposed into two components: one parallel to the ramp (mg sin θ) and one perpendicular to the ramp (mg cos θ). The normal force (N) is equal in magnitude but opposite in direction to the perpendicular component (mg cos θ) of the gravitational force.
No slipping occurs when the force of static friction exactly balances the component of gravity along the ramp. Therefore, we can set up the following equation:
μs * N = mg * sin θ
Substituting N = mg * cos θ, we get:
μs * mg * cos θ = mg * sin θ
Rearranging and canceling out the common terms, we find:
μs = tan θ
Plugging in the given angle θ of 50 degrees, we can compute the minimum coefficient of static friction:
μs = tan 50°
b) The minimum coefficient of static friction calculated in part (a) is specific to the solid ball and the angle of the ramp. It represents the critical value required to prevent slipping of the solid ball.
For a hollow ball (like a basketball or soccer ball), the mass distribution is different, and the surface area in contact with the ramp is much smaller compared to a solid ball. Therefore, a smaller coefficient of static friction would be sufficient to stop the hollow ball from slipping. In other words, the solid ball's coefficient of static friction would be more than enough to stop the hollow ball from slipping.
c) Part (a) asks about the coefficient of static friction because it is concerned with preventing slipping from occurring initially when the ball is at rest. When an object is at rest on an incline, the force opposing the potential sliding motion is labeled as static friction. Once the object starts sliding, the force opposing its motion is referred to as kinetic friction. In this scenario, the question is specific to the initial state of no slipping, so the coefficient of static friction is the required parameter to consider.