A cylinder and a cone are of same base radius and of same height. Find the ratio of the volume of the cylinder to the volume of the cone.

Explanation

(a) acceleration of cylinder parallel to slope (rolls up, so it is negative)
`a=-(gsintheta)/(1+(K^(2))/(R^(2)))`
If cylinder is solid, then radius of gyration
`K^(2)=(R^(2))/(2) [because" ring type"]`
`therefore a=-(9.8xxsin30^(@))/(1+(1)/(2))=-(2xx9.8xx(1)/(2))/(3)`
`therefore a=-(9.8)/(3)`
`therefore a=-3.267ms^(-2)`
At maximum height, velocity of cylinder `v=0`
`therefore` From equation of motion
`v^(2)-v_(0)^(2)=2ad`
`0-(5)^(2)=2(-3.267)d`
`therefore d=(25)/(2xx3.267)`
`therefore d=3.826~~3.83m`
when cylinder start to comes down with rolling `v_(0)=0,a=+3.267 ms^(-2) and d=3.83m`
`therefore` Equation of motion
`d=v_(0)t+(1)/(2)at^(2)`
`therefore 3.83=0+(1)/(2)xx3.267xxt^(2)`
`therefore (3.83xx2)/(3.267)=t^(2)`
`therefore t=sqrt((7.66)/(3.267))`
`therefore t=sqrt(2.34)`
`therefore t=1.53s`
Hence, time taken for go up and go down is same hence time taken to reached at bottom
`T=2t`
`=2xx1.53~~3.0s`