- #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:
$$mg\mu=mg\cos\theta~~\rightarrow~~\mu=\cos\theta$$