Question

A soild sphere is rolling on a frictionless plane surface about the axis of symmetry. Find the rotational energy of the sphere. Also, find the ratio of rotational `K.E.` to total energy.

Answer 1

Let `m` be the mass of soild sphere of radius `r`. Let `upsilon` be the uniform linear velocity of rolling.
Moment of inertia of the sphere, `I = (2)/(5)m r^(2)`
angular velocity pf shere, `omega = (upsilon)/(r )`
`:.` Rotational energy `= (1)/(2) I omega^(@)`
`= (1)/(2) xx (2)/(5) m r^(2) ((upsilon)/(r ))^(2) = (1)/(5)m upsilon^(2)`
Translational energy `= (1)/(2) m upsilon^(2)`
`:.` Total energy = Rotational energy
`+` Translational energy
`= (1)/(5) m upsilon^(2) + (1)/(2) m upsilon^(2) = (7)/(10) m upsilon^(2)`
`("Ratational energy")/("Total energy") = ((1)/(5)m upsilon^(2))/((7)/(10) m upsilon^(2)) = (2)/(7)`