A hollow sphere rolls down an incline. If it starts from rest, what fraction of

A hollow sphere rolls down an incline. If it starts from rest, what fraction of its total energy is translational at the bottom?

A hollow sphere rolls down an incline. If it starts from rest, what fraction of its total energy is translational at the bottom?

Step 1: Understand that the hollow sphere starts from rest at the top of the incline, meaning it has potential energy and no kinetic energy initially.
Step 2: As the hollow sphere rolls down the incline, its potential energy is converted into kinetic energy.
Step 3: Recognize that the total kinetic energy at the bottom consists of two parts: translational kinetic energy (movement of the center of mass) and rotational kinetic energy (spinning around its axis).
Step 4: For a hollow sphere, the relationship between translational kinetic energy (TKE) and rotational kinetic energy (RKE) is such that TKE is 2/3 of the total kinetic energy at the bottom.
Step 5: Therefore, if we denote the total kinetic energy at the bottom as KE_total, we can express TKE as TKE = (2/3) * KE_total.
Step 6: To find the fraction of the total energy that is translational, we divide the translational kinetic energy by the total energy: Fraction = TKE / Total Energy = (2/3) * KE_total / KE_total = 2/3.
Step 7: Conclude that the fraction of the total energy that is translational at the bottom of the incline is 2/3.

Conservation of Energy – The principle that the total energy in a closed system remains constant, allowing for the conversion between potential and kinetic energy.
Moment of Inertia – A measure of an object's resistance to changes in its rotation, which affects how energy is distributed between translational and rotational forms.
Rolling Motion – The combination of translational and rotational motion, where the total kinetic energy is the sum of translational kinetic energy and rotational kinetic energy.