What is the ratio of rotational kinetic energy for two balls, each tied to a light string and spinning in a circle with a radius equal to the length of the string?

Question

The first ball has a mass m attached to a string of length L, rotates at a rate of w and has rotational kinetic energy K_1rotational .The second ball has a mass 2m attached to a string of length 2L rotates at a rate of 2w and has rotational kinetic energy K_2rotational 

Enter answer as a fraction of integers.

Answer 1

The rotational kinetic energy is given by (0.5)(I)(ω²) where I is the moment of inertia and ω is the rotational velocity in radians per second.

For the first object I = m(L²) Therefore K1 =(0.5)mL²ω²

The second ball has a mass of 2m and L2=2L and ω2=2ω Therefore I = (0.5)(2m)(2L)² which gives us

K2 = (0.5)2m(2L)²(2ω)² = (16)mL²ω²

Therefore the ratio of the energies is 16/0.5 which means the second ball has 32 times more energy than the first ball