Yoyo linear velocity physics question. (energy method)?

Since the sphere is rotating as it falls to the ground, it has linear and rotational kinetic energy. Since we are using the energy method, the magnitude of its final linear and rotational kinetic energy will be equal to its initial potential energy.

PE = m * 9.8 * 1.2 = m * 11.76

Linear KE = ½ * m * v^2

Rotational KE = ½ * I * ω^2

For a sphere, I = 2/5 * m * r^2

Since we need to determine the final linear velocity, let’s convert ω to v by dividing by r.

ω^2 = v^2/r^2

Rotational KE = ½ * 2/5 * m * r^2 * v^2/r^2 = 0.2 * m * v^2

Total KE = 0.7 * m * v^2

Set this equal to the initial potential energy and solve for v.

0.7 * m * v^2 = m * 11.76

v^2 = 11.76 ÷ 0.7 = 16.8

v = √16.8

This is approximately 4.1 m/s.

It has energy mgh where h = 1.2 m.

This is turned into kinetic energy. There is linear kinetic energy (1/2)mv^2 and there is rotational kinetic energy (1/2) Iw^2 where w = angular velocity = v/r and I = moment of inertia of a sphere. Write down the formula for moment of inertia of a sphere. Add the two kinetic energies and set them equal to mgh. Solve for v.