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

Dr. Jie Zou PHY 1151G Department of Physics Torque Definition of torque, , for a tangential force: Magnitude of  :  = rF. SI units: Nm. Tangential force: the applied force is tangential to a circle of radius r centered on the axis of rotation. Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Exercise 11-1: To open the door in the figure a tangential force F is applied at a distance r from the axis of rotation. If the minimum torque required to open the door is 3.1 N·m, what force must be applied if r is (a) 0.94 m, or (b) 0.35 m? Dr. Jie Zou PHY 1151G Department of Physics

General definition of torque Radial force produces zero torque and thus causes no rotation. Radial force: force along a line that extends through the axis of rotation. It is the tangential component of the force alone that produces torque and thus causes rotation. General definition of torque, : Magnitude of  = r(F sin).  : the angle between the direction of the force and the radial direction. Dr. Jie Zou PHY 1151G Department of Physics

Sign convention for torque Sign convention for torque: By convention, if a torque  acts alone, then  >0 if the torque causes a counterclockwise angular acceleration.  <0 if the torque causes a clockwise angular acceleration. In a system of more than one torque, the sign of each torque is determined by the type of angular acceleration it alone would produce. Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Example 11-1 Two forces act on a wheel, as shown below. The wheel is free to rotate without friction, has a radius of 0.42 m, and is initially at rest. Given that F1 = 12 N and F2 = 9.5 N, find (a) the torque caused by F1 and (b) the torque caused by F2. (c) In which direction does the wheel turn as a result of these two forces? Dr. Jie Zou PHY 1151G Department of Physics

Torque and angular acceleration Newton’s second law for rotational motion:  = net = I ,  is the total (net) torque acting, I is the moment of inertia, and  is the angular acceleration caused by the torque. Analogies between rotational and linear quantities: Linear Quantity Angular Quantity m I a  F  Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Moment of inertia Definition of moment of inertia, I: I =  miri2. SI units: kg·m2. The precise value of I for a given object depends on its distribution of mass. Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Exercise 11-1: A light rope wrapped around a disk-shaped pulley is pulled with a force of 0.53 N. Find the angular acceleration of the pulley given that its mass is 1.3 kg and its radius is 0.11 m. (The moment of inertia of a disk rotating about the center axis is I = (1/2)mr2.) Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Static equilibrium Conditions for static equilibrium: (1) The net force acting on the object must be zero: Fx = 0, Fy = 0, and (2) The net torque acting on the object must be zero,  = 0. Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Active Example 11-2 A cat walking along a uniform plank that is 4.00 m long and has a mass of 7.00 kg. The plank is supported by two sawhorses, one 0.440 m from the left end of the board and the other 1.50 m from its right end. When the cat reaches the right end, the plank just begins to tip. What is the mass of the cat? Dr. Jie Zou PHY 1151G Department of Physics

Dr. Jie Zou PHY 1151G Department of Physics Homework Chapter 11, Page 341, Problems: #1, 8, 20, 35. Dr. Jie Zou PHY 1151G Department of Physics