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Torque and Newton’s Second Law Studio Physics I Part I-Basics of Torques Just as was the case for force, torque is a vector and so one must take its direction into account when applying Newton’s second law. Torques which act in opposite direction are assigned opposite signs (+ or -) and work against one another. Torques which act in the same direction are assigned the same sign (+ or -) and reinforce one another. Hence, we need to get good at determining the direction of a torque. Determining the direction of Torque using I To answer questions regarding direction, DO NOT USE + OR – unless you have clearly defined what direction you mean when you say + or -. Instead, try to use words like clockwise, counter clockwise, up, down, right, left, into the page or out of the page. 1. What is the direction of the angular velocity of the disk shown at the left? (the axis points out of the page) 2. Do all points on the disk have the same angular velocity? Do all points on the disk have the same linear velocity? 3. What is the direction of the angular acceleration of this disk if it is speeding up? 4. What would the direction be of the torque necessary to make this disk spin in the direction shown with an increasing angular speed? 5. What is the direction of the angular acceleration of this disk if it is slowing down? 6. What would the direction be of the torque necessary to make this disk spin in the direction shown with an decreasing angular speed? 7. What is the direction of the angular velocity of the disk shown at the left? 8. What is the direction of the angular acceleration of this disk if it is speeding up? 9. What would the direction be of the torque necessary to make this disk spin in the direction shown with an increasing angular speed? 10. Consider the pulley shown at the right. There is a mass attached to a rope which is wrapped around the pulley. If the mass is let free to fall, what is the direction of the angular velocity of the pulley? What is the direction of the angular acceleration? What is the direction of the torque that the mass and rope exert on the pulley? COPYRIGHT K. Cummings 1999, 2000 Determining the direction of Torque using r F To answer questions regarding direction, DO NOT USE + OR – unless you have clearly defined what direction you mean when you say + or -. Instead, use words like up, down, right, left, into the page or out of the page. 11. Use the right hand rule to state the direction of c ab for each case shown below. (Hint: the answer to A is out of the page) A) B) C) D) a b a a b a b b 12. Use the right hand rule to state the direction of c ba for each case shown above. What is the relationship between ba and ab? 13. Consider the pulley shown at the left There is a mass attached to a rope which is wrapped around the pulley. If the mass is let free to fall, what is the direction of the tension in the string? Draw a set of two arrows, one which represents the direction of the tension in the string and one which represents the radius of the pulley. The direction of the radius is determined by drawing an arrow from the center of the pulley to the point at which the force (tension in this case) is applied. Use these two arrows and rF to determine the direction of the torque produced by the tension. Compare your answer to your answer in question # 10 above. Applying Newton’s Second Law with Torques Newton’s second law F=ma can be just as useful for analyzing rotational motion as it was for translational motion. However, in analyzing rotational motion, it is the sum of the torques acting on the object that determines what the angular acceleration on the object will be. In other words, F=ma becomes = . . For objects in equilibrium (that is, stationary, balanced objects) both F=0 and =0 14. COPYRIGHT K. Cummings 1999, 2000 COPYRIGHT K. Cummings 1999, 2000 Part II-Designing and Building a Mobile COPYRIGHT K. Cummings 1999, 2000 Part III Angular Momentum A student is standing on a frictionless turntable that is free to rotate about a vertical axis through the student's center of mass. The rotational inertia of the student and turntable, about the axis of rotation, is I = 3.0 kg·m². The student and turntable are initially at rest. A second student throws a ball of mass 4.0 kg at a speed of 4.0 m/s which the student on the turntable catches. When the ball is caught, it is 0.5 m distant from the axis of rotation, as shown in the sketch. 1) Find the angular momentum of the ball about the axis of rotation of the turntable before the ball is caught. 2) Find the angular velocity of the student plus turntable plus ball after the ball has been caught. 3) Find the kinetic energy of the ball before it is caught. 4) Find the kinetic energy of the student plus turntable plus ball after the ball has been caught. COPYRIGHT K. Cummings 1999, 2000