A uniform ball of radius `r` rolls without slipping down from the top of a sphere of radius `R` Find the angular velocity of the ball at the moment it breaks off the sphere. The initial velocity of the ball is negligible.

Options

A. `sqrt((5g(R+r))(17r^(2)))`
B. `sqrt((10g(R+r))/(17r^(2)))`
C. `sqrt((5g(R-r))/(10r^(2)))`
D. `sqrt((10g(R+r))/(7r^(2)))`

Answer

B

Explanation

`(mv^(2))/((R+r))=mgcostheta`
`mgh=l1/2mv^(2)+1/2Iomega^(2)`

`mg(R+r)(1-costheta)=1/2mv^(2)+1/5mv^(2)`
`=7/10mv^(2)`
`10/7mg(R+r)(1-costheta)=mgcostheta`
`mv^(2)=10/7mg(R+r)(1-costheta)`
`10/7=17/7costheta or costheta=10/7`
`v=sqrt(g(R+r)costheta)=sqrt(10/7g(R+r))`
and `omega=v/r=sqrt((10g(R+r))/(17r^(2)))`