A solid sphere of mass `m` and radius `R` rolls without slipping on the horizontal surface such that its velocity of center of mass is `v_(0)`. Find its translational kinetic energy, rotational kinetic energy, total kinetic energy, gravitational potential energy taking surface as reference level and mechanical energy.

Explanation

As the body rolls without slipping it has linear as well as angular velocity such that `v_(c.m.)=v_(0)=R omega`
`K_(T)=(1)/(2)mv_(c.m.)^(2)=(1)/(2)mv_(0)^(2)`
`K_(R)=(1)/(2)I_(c.m.)omega^(2)=(1)/(2)xx(2)/(5)mR^(2)((v_(0))/(R))^(2)=(1)/(5)mv_(0)^(2)`
`K=K_(T)+K_(R)=((1)/(2)+(1)/(5))mv_(0)^(2)=(7)/(10)mv_(0)^(2)`
`U=mgR`
Mechanical energy `E=K+U=(7)/(10)mv_(0)^(2)+mgR`
Note: for quick calculation, use
`K_(R)=(k^(2))/(R^(2))K_(T)=(k^(2))/(R^(2))(1)/(2)mv_(c.m.)^(2)=(2)/(5)xx(1)/(2)mv_(0)^(2)=(1)/(5)mv_(0)^(2)`
[for solid sphere `(k^(2))/(R^(2))=(2)/(5)`]
`K=(1+(k^(2))/(R^(2)))(1)/(2)mv_(c.m.)^(2)=(1+(2)/(5))(1)/(2)mv_(0)^(2)=(7)/(10)mv_(0)^(2)`