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ENGR 215 ~ Dynamics Sections 15.5 – 15.7 Angular Momentum • “Moment” of the particle’s linear momentum about Point O. • Scalar Formulation – If the particle is moving along a curve lying in the x-y plane. H O z (d )(mv) Angular Momentum • Vector Formulation – If the particle is moving along a space curve. H O r mv Lecture Example 1: Determine the angular momentum HO of the particle about Point O. Relation between Moment of a Force and Angular Momentum Relation between Moment of a Force and Angular Momentum • The time rate of change of a particle’s angular momentum is equal to the torque acting on it. M H O O • The time rate of change of a particle’s linear momentum is equal to the force acting on it. F L Angular Impulse & Momentum Principles dH O M O H O dt M O dt dH O t2 M O dt (H O ) 2 (H O )1 t1 t2 (H O )1 M O dt (H O ) 2 t1 t2 Angular Impulse M O dt t1 t2 (r F ) dt t1 Linear and Angular Impulse t2 mv1 Fdt mv 2 t1 t2 (H O )1 M O dt (H O ) 2 t1 Conservation of Angular Momentum • When the angular impulses acting on a particle are all zero during a time from t1 to t2, angular momentum is conserved. (H O )1 (H O ) 2 Lecture Example 2: A 4-lb ball is traveling around a circle of radius r = 3 ft with a speed (vb)1 = 6 ft/s. If the cord is pulled through the hole with a constant speed, determine the speed of the ball when r = 2 ft. Also, how much work had to be done to pull down the cord? Lecture Example 3: The 800-lb roller-coaster starts from rest on the track having the shape of a cylindrical helix. It the helix descends 8 ft for every revolution, determine the time required for the car to attain a speed of 60 ft/s. Neglect friction and the size of the car,