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) Let us indicate the forces acting on the sphere and their points of application. Choose positive direction of x and `varphi` (rotation angle) along the incline in downward direction and in the sense of `vecomega` (for undirectional rotation) respectively. Now from equations of dynamics of rigid body i.e. `F_x=mw_(cx)` and `N_(cz)=I_cbeta_z` we get:
`mgsinalpha-f_r=mw` (1)
and `f rR=2/5mR^2beta` (2)
But `f rlekmgcosalpha` (3)
In addition, the absence of slipping provides the kinematical relationship between the accelerations:
`w=betaR` (4)
The simultaneous solution of all the four equations yields:
`kcosalphage2/7sinalpha`, or `kge2/7tanalpha`
(b) Solving Eqs. (1) and (2) [of part (a)], we get:
`w_c=5/7g sin alpha`.
As the sphere starts at `t=0` along positive x axis, for pure rolling
`v_c(t)=w_ct=5/7gsinalphat` (5)
Hence the sought kinetic energy
`T=1/2mv_c^2+1/22/5mR^2omega^2=7/10mv_c^2` (as `omega=v_c//R`)
`=7/10m(5/7gsinalphat)^2=(5)/(14)mg^2sin^2alphat^2`