Work Let us examine the work done by a torque applied to a system. This is a small amount of the total work done by a torque to move an object a small angular distance d . Power Similar to Work – Kinetic Energy

Rolling motion (No Slipping) The majority of motion we have been discussing is translational motion. We have recently been exploring rotational motion. Now we will look at both together. P P’ v = v CM v = 2v CM v = 0 If a wheel is placed on a flat surface and a force is applied at the center of the wheel what will it do? It will translate and rotate. Why does it rotate? It rotates due to the frictional force at the point of contact, that is in a direction opposite to the direction the wheel would slip. P f The rolling motion associated with this wheel can be modeled as if all parts of the wheel rotate about the point of contact. For this instant in time the point of contact would have zero velocity, the center of mass would have a velocity v CM and the top of the wheel would have a velocity twice that of the center of mass. Using this model, what can we say about the velocities of point P, the center and point p’? v CM 

S S R How can we determine the translational velocity of the center of mass when we only know the rotational speed of the wheel?   We can look at the translational distance covered by the wheel. The angular velocity of the center about the contact point is the same as the angular velocity of the contact point around the center. We can also look at the acceleration of the center of mass. Let us also look at the kinetic energy associated with rolling. Rotational Energy Translational energy

Angular Momentum Angular momentum describes the momentum of an object as it rotates about an axis. This term is introduced to arrive at the correct result. The end result is known and therefore this result is more obvious when working the derivation in the opposite direction. We are adding 0 to both sides. This resembles the impulse- momentum equation. L – Angular momentum [kg m 2 /s] Impulse-angular momentum equation A net torque causes a change in the angular momentum of a system. This describes the angular momentum of a single particle. This describes the angular momentum of a single particle, where r and v are perpendicular. This would describe a particle moving in a circle.

If we want to look at a collection of particles, we can find the total angular momentum by summing the angular momentums of each particle. Relates the angular momentum to the angular velocity for the i th particle. Angular momentum of a rigid object (continuous body). Original definition of torque. If there is no net torque on the system: L is constant Angular momentum is conserved if there is no net torque on a system.