Rotational Dynamics and Static Equilibrium (Cont.) Chapter 11 Rotational Dynamics and Static Equilibrium (Cont.) Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Outline Dynamic Applications of Torque Applications of Newton’s 2nd law to rotations Angular Momentum Conservation of Angular Momentum Dr. Jie Zou PHY 1151G Department of Physics

Dynamic applications of torque Application of Newton’s second law for rotational motion: net = I net : the net torque (or the total torque, ); I: the moment of inertia; : the angular acceleration Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Example 1 Consider a disk-shaped pulley of radius R and mass M with a string wrapped around its circumference. Hanging from the string is a mass m. When the mass is released, it accelerates downward and the pulley begins to rotate. If the pulley rotates without friction, the string unwraps without slipping, and the pulley has a moment of inertia of I, what is the acceleration of the mass and the tension in the string? Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Example 2 A 0.31-kg cart on a horizontal air track is attached to a string. The string passes over a disk-shaped pulley of mass 0.080 kg and radius 0.012 m and is pulled vertically downward with a constant force of 1.1 N. Find (a) the tension in the string between the pulley and the cart, and (b) the acceleration of the cart. Answers: (a) 0.97 N; (b) 3.1 m/s2 Note: when the moment of inertia of the pulley cannot be neglected, T1 ≠ T2. Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Angular Momentum Definition of angular momentum, L = I. SI unit: kg·m2/s. Exercise: Find the angular momentum of a 0.1-kg Frisbee (considered to be a uniform disk of radius 7.5 cm) spinning with an angular speed of 1.15 rad/s. Dr. Jie Zou PHY 1151G Department of Physics

Conservation of angular momentum If the net external torque acting on a system of objects is zero, the angular momentum is conserved: Lf = Li Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Example A student sits on a piano stool holding a sizable mass in each hand. Initially, the student holds his arms outstretched and spins about the axis of the stool with an angular speed of 3.72 rad/s. The moment of inertia in this case is 5.33 kg·m2. If the student pulls his arms in to his chest, reducing the moment of inertia to 1.60 kg·m2. What is the student’s angular speed now? Answer: 12.4 rad/s Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Homework See online homework on www.masteringphysics.com Hand-written problems: Ch. 11, p. 369, Problems: #46, 47. Dr. Jie Zou PHY 1151G Department of Physics