A small solid sphere of radius `r` rolls down an incline without slipping which ends into a vertical loop of radius `R`. Find the height above the base so that it just loops the loop

Options

A. `5/2R`
B. `5/2(R-r)`
C. `25/10(R-r)`
D. `27/10R-(17r)/10`

Answer

D

Explanation

The minimum velolcity of `P`, topo of the loop, should be `v=sqrt(g(R-r))` if the sphere keeps on rolling at top `v=omegaR`

`mgh=1/2mv^(2)+1/2Iomega^(2)+mg(2R-r)`
`=1/2mg(R-r)+1/2(2/5) mR^(2)2omega^(2)+mg(2R-r)`
`=7/10 mg(R-r)+mg(2R-r)[omegar=v=sqrt(g(R-r))]`
`=mg/10(27R-17r)or h=1/10(27r- 17r)`