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A uniform sphere of mass m and radius r ...

A uniform sphere of mass `m` and radius `r` rolls without sliding over a horizontal plane, rotating ahout a horizontal axle `OA`. In the process, the centre of the sphere moves with a veocity `v` along a circle of radius `R`. Find the kinetic energy of the sphere.

Text Solution

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Here the motion of the body is oribital and spin combined. The sphere is spining about a horizontal axis and orbiting about a vertical axis.
We write total kinetic energy of the sphere i.e.,
`KE=K_("orbital")+K_("spin")`

The kinetic energy of the sphere due to its rotation about its own axis nad its motion along circular path with velocity `v`.
`KE=1/2I_(1)omega_(1)^(2)+1/2I_(2)omega_(2)^(2)`
where `I_(1)=2/5mr^(2)` and `I_(2)=(2/5mr^(2)+mR^(2))`
`omega_(1)=v/r` and `omega_(2)=v/R`
Substituting these values in the above equation, we get
`KE=(7m)/10 v^(2)[1+(2r^(2))/(7R^(2))]`
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