Question

a solid sphere is rotating about an axis that passes through its center, and so is a thin-walled spherical shell. they have identical masses, radii, and kinetic energies. the angular speed of the shell is 6.81rad/s. what is the angular speed of the solid sphere?

Answer 1

The angular speed of the solid sphere can be found by setting the kinetic energies equal for both the solid sphere and the thin-walled spherical shell, and solving for the unknown angular speed, using their respective moments of inertia.

Step-by-step explanation:

The question involves comparing the angular speeds of a solid sphere and a thin-walled spherical shell that have identical masses, radii, and kinetic energies. The moment of inertia (I) for a solid sphere is ½MR² and for a thin-walled spherical shell, it is 2⁄3MR². Since the kinetic energy (KE) of rotation for both objects is the same and given by ½Iω², where ω is the angular speed, we can set up an equation relating the kinetic energies of the two objects for comparison.

Since the shell rotates at an angular speed of 6.81 rad/s, we can use the equation KE = ½I_shellω_shell² = ½I_sphereω_sphere² to find the angular speed of the sphere. After working through the algebra and considering the different moments of inertia, we arrive at ω_sphere = ω_shell∙(∙⁄3⁄2), which when calculated with the provided speed of the shell, yields the angular speed of the solid sphere.