A sphere of radius 2.84 cm and a cylinder of radius 8.97 cm are rolling without slipping along the same floor. The two objects have the same mass. If they are to have the same total kinetic energy, what should the ratio of the sphere\'s angular speed to the cylinder\'s angular speed be?

To find the ratio of the sphere's angular speed to the cylinder's angular speed, we can start by looking at the formula for the kinetic energy of a rotating object.

The kinetic energy of a rotating object is given by the formula:

KE = (1/2) * I * ω^2

Where KE is the kinetic energy, I is the moment of inertia, and ω is the angular speed.

For a sphere, the moment of inertia is given by:

I_sphere = (2/5) * m * r^2

Where m is the mass of the sphere and r is the radius of the sphere.

For a cylinder, the moment of inertia is given by:

I_cylinder = (1/2) * m * r^2

Where m is the mass of the cylinder and r is the radius of the cylinder.

Since the mass of both the sphere and the cylinder is the same, we can cancel it out when finding the ratio of their kinetic energies.

Setting the kinetic energies of the sphere and the cylinder equal to each other, we can solve for the ratio of their angular speeds:

(1/2) * (2/5) * r^2 * ω_sphere^2 = (1/2) * (1/2) * r^2 * ω_cylinder^2

Simplifying the equation, we get:

(2/5) * ω_sphere^2 = (1/4) * ω_cylinder^2

To find the ratio of the sphere's angular speed to the cylinder's angular speed, we divide both sides of the equation by ω_cylinder^2:

(2/5) * (ω_sphere^2 / ω_cylinder^2) = 1/4

The left side of the equation simplifies to:

(2/5) * (ω_sphere / ω_cylinder)^2 = 1/4

Taking the square root of both sides of the equation allows us to solve for the ratio:

(2/5) * (ω_sphere / ω_cylinder) = 1/2

Dividing both sides of the equation by (2/5), we get:

ω_sphere / ω_cylinder = 1/5

Therefore, the ratio of the sphere's angular speed to the cylinder's angular speed should be 1/5.