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Work Let us examine the work done by a torque applied to a system. dW F ds Ft ds F sin ds F sin rd dW d Power This is a small amount of the total work done by a torque to move an object a small angular distance d. dW d d P dt dt dt P Similar to P F v Work – Kinetic Energy d d d d I I dt I d dt I d d Id 1 2 1 2 W Id W I I0 K R 2 2 0 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. 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. The rolling motion associated with this wheel can be modeled as if all parts of the wheel rotate about the point of contact. Using this model, what can we say about the velocities of point P, the center and point p’? For this instant in time the point of contact would have zero velocity, the center of mass would have a velocity vCM and the top of the wheel would have a velocity twice that of the center of mass. P’ v = 2vCM vCM v = vCM v=0 P f P 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. vcm ds Rd R dt dt 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. acm S R dvCM Rd R dt dt S Let us also look at the kinetic energy associated with rolling. 1 1 1 2 1 2 2 2 K I I CM MD I CM MR 2 2 2 2 2 2 1 1 2 2 K I CM MvCM Translational energy 2 2 Rotational Energy Angular Momentum Angular momentum describes the momentum of an object as it rotates about an axis. This term is introduced to arrive at dp dp dr the correct result. The end result is r F r r p known and therefore this result is dt dt dt more obvious when working the d r p This resembles the impulsederivation in the opposite direction. momentum equation. dt dr v dt v mv 0 L r p L – Angular momentum [kg m2/s] We are adding 0 to both sides. dL Impulse-angular A net torque causes a change in the dt momentum equation angular momentum of a system. L r p rmv sin This describes the angular momentum of a single particle . L rmv 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. Li ri mi vi ri mi ri ri 2 mi Relates the angular momentum to the angular velocity for the ith particle. 2 L Li ri mi L I i i dL d I d dt dt I dt I Angular momentum of a rigid object (continuous body). Original definition of torque. If there is no net torque on the system: dL dt 0 Li L f dL 0 L is constant L 0 Angular momentum is conserved if there is no net torque on a system.