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

(i) If a body rolls without slipping
`mu_(min)=(tan theta)/(I+(R^(2))/(v^(2)))=(tan theta)/(1+(5)/(2))=(2)/(7) tan theta`
(ii) For pure rolling
`a=(g sin theta)/(1+(k^(2))/(R^(2)))=(g sin theta)/(1+(2)/(5))=(5)/(7) g sin theta`
Linear velocity after time `t`
`v=u+at=0+(5g sin thetat)/(7)`
In pure rolling KE
`K=(1)/(2)mv^(2)(1+(k^(2))/(R^(2)))=(1)/(2)mv^(2)(1+(2)/(5))`
`=(7)/(10)mv^(2)=(7)/(10)m((5g sin theta t)/(7))^(2)=(5)/(14)mg^(2)sin^(2) thetat^(2)`.