A uniform sphere of mass `m` and radius `R` rolls without slipping down an inclined plane set at an angle `theta` to the horizontal. Find
(`a`) the friction coefficient at which slipping is absent,
(`b`) the kinetic energy of the sphere `t` seconds after the beginning of motion.

Explanation

(`a`) If a body rolls without slipping
`mu_(min)=(tantheta)/(1+(R^(2))/(k^(2)))`(for solid sphere, `(k^(2))/(R^(2))=(2)/(5)`)
`=(tantheta)/(1+(5)/(2))`
`=(2)/(7)tantheta`
(`b`) For pure rolling,
`a=(gsintheta)/(1+(k^(2))/(R^(2)))=(gsintheta)/(1+(2)/(5))=(5)/(7)g sintheta`
Linear velocity after time `t`
`v=u+at=0+(5gsintheta t)/(7)`
In pure rolling `K.E.`
`K=(1)/(2)mv^(2)(1+(k^(2))/(R^(2)))`
`=(1)/(2)mv^(2)(1+(2)/(5))`
`=(7)/(10)mv^(2)=(7)/(10)m((5gsintheta t)/(7))^(2)`
`=(5)/(14)mg^(2)sin^(2)theta t^(2)`