A ring is rolling on a horizontal surface without slipping. Find the ratio of translation kinetic energy to rotational kinetic energy of ring.

Let m be the mass and r be the radius of ring.

When the ring rolls without slipping, then v = rω,

where v is the velocity of centre of mass of ring, and
ω is its angular velocity.

Now the translational kinetic energy of the ring is,

$straight K subscript straight t space equals space 1 half mv squared space space space space space equals 1 half mr squared straight ϖ squared space$

Rotational kinetic energy of the ring is,

$straight K subscript straight r space equals 1 half Iϖ to the power of 2 space end exponent space space space space space equals space 1 half mr squared straight ϖ squared Therefore comma space the space ratio space o f space translational space to space rotational straight K. straight E space is space given space by comma space therefore space space straight K subscript t over straight K subscript r equals 1$

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