Rotational Equilibrium and Dynamics Rotational Energy & Rotational Dynamics Problems

Rotational Kinetic Energy You have learned previously that the mechanical energy of an object includes its translational kinetic energy and its potential energy. This approach did not consider the possibility that objects could have rotational motion along with translational motion.

Rotational Kinetic Energy Rotating objects possess kinetic energy associated with their angular speed. We call this energy rotational kinetic energy. (rotational kinetic energy)=1/2(moment of inertia)x(angular speed)2

Mechanical Energy Taking rotational motion into account, the mechanical energy of a system is the sum total of the translational kinetic energy, gravitational potential energy, and rotational kinetic energy.

Conservation of Mechanical Energy Recall that the total energy of a system is conserved.

Analysis of the rolling cans: Racing identical pop cans. What happens? What energies do the cans have at the top of the ramp? At the bottom? Which has more translational KE? Which has more rotational KE?

Linear v. Angular Quantities Displacement x q Velocity v w Acceleration a Inertia m I Momentum p=mv L=Iw Kinetic Energy ½ m v2 ½ Iw2

Sample Problem #1 A solid ball with a mass of 4.10 kg and a radius of 0.050 m starts from rest at a height of 2.00 m and rolls down a 30.0o slope. What is the translational speed of the ball when it leaves the incline?

Sample Problem #2 As Halley’s comet orbits the sun, its distance from the sun changes dramatically, from 8.8x1010 m to 5.2x1012 m. If the comet’s speed at closest approach is 5.4x104 m/s, what is its speed when it is farthest from the sun if angular momentum is conserved?

Sample Problem #3 Assume that a yo-yo has a mass of 6.00x10-2 kg. If a yo-yo descends from a height of 0.600 m down a vertical string and had a linear speed of 1.80 m/s by the time it reached the bottom of the string. If its final angular speed was 82.6 rad/s, what was the yo-yo’s moment of inertia?