A hollow sphere is released from the top of an inlcined plane of inclination `theta`. A. What should be the minimum coefficientof friction between the shphere and the plane to prevent sliding? B. Find the kinetic energy of the ball as it moves down a length l on the incline if the friction coefficient is half the value calculated in part a.

Answer

A, B

Explanation

A hollow sphere is released from a top of an inclined plane a of incliN/Ation `theta`.
a. To prevent slidint the body will make only perfect rolling In this condition
`mglsintheta-f=ma` …………..1
`fxxR=(2/3)mR^2x(a/R)`
`rarr f=2/3ma`…………2
`rarr putting this value in the equation 1 we get
`- mg sintheta-2/3 ma=ma`

`rarr a=3/5a sin theta`
rarr mgsintheta-f=3/5mgsintheta`
`rarr f=2/5mgsintheta`
`rarr m mgcostheta-2/5 mg sintheta`
`rarrmu=2/5 tan theta`
`b. (1/5) tantheta(mgcostheta)R=2/3mR^2alpha`
`rarr alpha=3/10 ((gsintheta)/R)`
`ac=g sintheta-(g/5)sintheta`
`rarr =(4/5)gsintheta`
`rarr t^2=(2s)/(ac)`
`=2l(4g(sintheta)/5)(5/(2gsintheta))`
`:. Agasin, omega=alphat`
`K.E. =1/2mv^2+1/2Iomega^2`
`=1/2m(2as)+1/2lmv^2+1/2Iomega^`
`=1/2(2as)+1/2l(alpha^2t^2)`
`=1/2m(4g(sintheta)/5)x2xxl+1/2x2/3mR^2x9/100`
`((sin^2theta)/R)xx((5L)/(2gsintheta))`
`-4mgl(sintheta)/5+3mgl(sintheta)/40`
`=7/8mglsintheta`