A solid sphere of mass `m` rolls without slipping on an inclined plain of inclination `theta`. What should be the minimum coefficient of static friction to support pure rolling?

Options

A. `tan theta ge 3mu_(s)`
B. `tan theta gt 3 mu_(s)`
C. `tan theta le 3 mu_(s)`
D. `tan theta lt 3 mu_(s)`

Answer

C

Explanation

`a= (g sin theta)/((1+ ""K^(2))//_(R^(2)))`
`a= (2)/(3) g sin theta` [As `(K^(2))/(R^(2))= (1)/(2)`]
The torque
`tau= f R= I alpha`
`rArr f R= (MR^(2))/(2) alpha`
Since, `R alpha= a rArr f R= (MR)/(2) a`
`rArr f= (Ma)/(2)`
`f= (M)/(2).(2)/(3) g sin theta`
`rArr mu_(s) mg cos theta= (m)/(3) g sin theta rArr mu_(s)= (tan theta)/(3)`
For rolling without slipping `tan theta le 3 mu_(s)`