A solid cylinder rolls down an incline of height h. What fraction of its total m

A solid cylinder rolls down an incline of height h. What fraction of its total mechanical energy is kinetic energy at the bottom?

A solid cylinder rolls down an incline of height h. What fraction of its total mechanical energy is kinetic energy at the bottom?

Step 1: Understand that a solid cylinder rolling down an incline starts with potential energy due to its height (h).
Step 2: Recognize that as the cylinder rolls down, this potential energy converts into kinetic energy.
Step 3: Know that the total kinetic energy at the bottom consists of two parts: translational kinetic energy (due to its linear motion) and rotational kinetic energy (due to its spinning).
Step 4: For a solid cylinder, the formula for total kinetic energy at the bottom is: Kinetic Energy = Translational KE + Rotational KE.
Step 5: The translational kinetic energy is given by (1/2)mv^2 and the rotational kinetic energy is given by (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Step 6: For a solid cylinder, the moment of inertia I = (1/2)mr^2 and the relationship between linear velocity (v) and angular velocity (ω) is v = rω.
Step 7: Substitute the values into the kinetic energy formulas to find the total kinetic energy.
Step 8: After calculations, determine that 2/3 of the total mechanical energy at the bottom is kinetic energy.

Conservation of Energy – The principle that total mechanical energy (potential + kinetic) remains constant in the absence of non-conservative forces.
Kinetic Energy Distribution – Understanding how total kinetic energy is divided into translational and rotational components for rolling objects.
Moment of Inertia – The role of the moment of inertia in determining the distribution of energy between translational and rotational motion.