# Ball down hill w/ rotational motion

• QuarkCharmer

## Homework Statement

A solid ball is released from rest and slides down a hillside that slopes downward at an angle 69.0 degrees from the horizontal.

What minimum value must the coefficient of static friction between the hill and ball surfaces have for no slipping to occur?

## The Attempt at a Solution

I'm not really sure what they mean here. If the ball is to roll down the hill, then it cannot "slide" at all, otherwise it would not roll. They don't give me any numbers to solve this with, and it's not a symbolic answer because the problem does not say "give your solution in terms of m,g,θ, et al.

What do they want me to do for this problem?

What I did was treat the ball as a box, and find the $μ_{s}$ like so:

I said that parallel to, and down the hill was the increasing x axis. Then I summed up the forces, applied F=ma, so solve:

$$mgsin(69)-f_{s}=ma$$
$$f_{s} = μ_{s}N = μ_{s}(mgcos(69))$$
$$mgsin(69)-μ_{s}(mgcos(69)=ma$$
Because it's not slipping, acceleration is zero, thus ma = 0
$$mgsin(69)-μ_{s}(mgcos(69)=0$$
$$mgsin(69) =μ_{s}(mgcos(69)$$
$$μ_{s} = \frac{mgsin(69)}{mgcos(69)}$$
$$μ_{s} = tan(69)$$

Does that make sense?

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