- #1

Karol

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## Homework Statement

The ball of mass m and radius r is released from horizontal position, the edge of the bowl.

What are the tangential and radial accelerations at angle θ and what is the velocity.

What should be the minimal friction coefficient at angle θ so that the ball won't slip

## Homework Equations

Radial acceleration: ##a_r=\frac{v^2}{r}##

## The Attempt at a Solution

Conservation of energy: ##mgh=\frac{1}{2}mv^2~~\rightarrow~~mg(R-r)\sin\theta=\frac{1}{2}mv^2##

$$v^2=2g(R-r)\sin\theta$$

$$a_t=\frac{mg\cos\theta}{m}=g\cos\theta$$

$$a_r=\frac{v^2}{r}=\frac{2g(R-r)\sin\theta}{R}$$

In order not to slip the tangential forces of friction and the component of gravity must cancel:

$$mg\mu=mg\cos\theta~~\rightarrow~~\mu=\cos\theta$$