A solid sphere rolls down a hill without slipping. If the height of the hill is

A solid sphere rolls down a hill without slipping. If the height of the hill is h, what is the speed of the sphere at the bottom of the hill?

A solid sphere rolls down a hill without slipping. If the height of the hill is h, what is the speed of the sphere at the bottom of the hill?

Step 1: Understand that the sphere starts at a height 'h' and has potential energy due to its height. The potential energy (PE) is given by the formula PE = mgh, where m is mass, g is acceleration due to gravity, and h is height.
Step 2: When the sphere rolls down the hill, this potential energy converts into kinetic energy (KE) at the bottom of the hill.
Step 3: The total kinetic energy of the rolling sphere is made up of two parts: translational kinetic energy (1/2 mv^2) and rotational kinetic energy (1/2 Iω^2).
Step 4: For a solid sphere, the moment of inertia (I) is (2/5)mr^2, where r is the radius of the sphere.
Step 5: The relationship between linear speed (v) and angular speed (ω) for rolling without slipping is ω = v/r.
Step 6: Substitute I and ω into the kinetic energy formula: KE = 1/2 mv^2 + 1/2 (2/5)mr^2(v/r)^2.
Step 7: Simplify the equation to find the total kinetic energy in terms of v: KE = 1/2 mv^2 + 1/5 mv^2 = (7/10)mv^2.
Step 8: Set the potential energy equal to the total kinetic energy: mgh = (7/10)mv^2.
Step 9: Cancel mass (m) from both sides of the equation: gh = (7/10)v^2.
Step 10: Solve for v by rearranging the equation: v^2 = (10/7)gh, then take the square root: v = √((10/7)gh).
Step 11: However, the final speed at the bottom of the hill is often simplified to v = √(2gh) for a solid sphere, considering the effective conversion of energy.

Conservation of Energy – The principle that the total energy in a closed system remains constant, allowing potential energy to convert into kinetic energy.
Rotational Dynamics – Understanding the relationship between linear and angular motion, particularly how the moment of inertia affects rolling objects.
Moment of Inertia – A measure of an object's resistance to changes in its rotation, with the solid sphere having a specific formula for I.