LECTURE 24 ANGULAR MOMENTUM Instructor: Kazumi Tolich Lecture 24 2 ¨ Reading chapter 11-6 ¤ Angular momentum n Angular momentum about an axis n Newton’s 2nd law for rotational motion Angular momentum of an rotating object 3 ¨ An object with a moment of inertia of � about an axis rotates with an angular speed of � about the same axis has an angular momentum, �, given by � = �� ¤ This is analogous to linear momentum: � = �� Angular momentum in general 4 ¨ Angular momentum of a point particle about an axis is defined by � � = �� sin � = ��� sin � = �-� = ��. � �- ¤ �⃗: position vector for the particle from the axis. ¤ �: linear momentum of the particle: � = �� �⃗ ¤ � is moment arm, or sometimes called “perpendicular . Axis distance.” �. Quiz: 1 5 ¨ A particle is traveling in straight line path as shown in Case A and Case B. In which case(s) does the blue particle have non-zero angular momentum about the axis indicated by the red cross? A. Only Case A Case A B. Only Case B C. Neither D. Both Case B Quiz: 24-1 answer 6 ¨ Only Case A ¨ For a particle to have angular momentum about an axis, it does not have to be Case A moving in a circle. ¨ The particle can be moving in a straight path. Case B ¨ For it to have a non-zero angular momentum, its line of path is displaced from the axis about which the angular momentum is calculated. ¨ An object moving in a straight line that does not go through the axis of rotation has an angular position that changes with time. So, this object has an angular momentum. ¨ � = ��. Sign convention of L 7 ¨ The sign of � is determined by the angle to the object. ¤ Positive � for increasing �: an object in (a) or an object moving counterclockwise in (b). ¤ Negative � for decreasing � : opposite motions from (a) or (b). Example: 1 8 ¨ You observe a particle with a mass � = 2.0 kg moving at a constant speed of � = 3.5 m/s in a clockwise direction around a circle of radius � = 4.0 m. a) What is its angular momentum (including the sign) about the center of the circle? b) What is its moment of inertia about an axis through the center of the circle and perpendicular to the plane of the motion? c) What is the angular speed of the particle? Total angular momentum of a system of particles 9 ¨ The total angular momentum of a system of particles are the sum of the angular momentum of the individual particles: 7 �-4- = 5 �6 7 ¨ You need to include the sign of angular momentum of each particle. Quiz: 2 ¨ Particles 1, 2, and 3 have equal masses and equal speeds. Rank the particle according to the magnitude of the angular momentum with respect to the origin, smallest first. Velocity vectors are shown. Quiz: 24-2 answer ¨ �8 < �: < �; ¨ Since the particles have equal masses and speeds, the magnitudes of the linear momenta are the same. ¨ So, the ranking of the magnitude of angular momentum is determined by the ranking of the moment arm: � = ��. ¤ Particle 1: 1 unit ¤ Particle 2: 3 units ¤ Particle 3: 2 units Velocity vectors are shown. Quiz: 3 ¨ The particles are not accelerating. How does the angular momentum of particle 1 change with time? A. Increases with time B. Decreases with time C. Stays the same Velocity vectors are shown. Quiz: 30-3 answer ¨ Stays the same. ¨ Since particle 1 is not accelerating, the velocity remains the same. ¨ Since the mass does not change, the linear momentum of particle 1 does not change. ¨ The moment arm also remains the same as 1 unit long. ¨ � = ��. Velocity vectors are shown. Newton’s 2nd law for angular motion 14 ¨ The net external torque applied to a system about an axis (the system is non-isolated) equals the time rate of change of the angular momentum about that axis. 7 ∆� 5 � = ∆� 7 nd ¨ This is analogous to Newton’s 2 law for linear motion using linear ∆� momentum, ∑7 �⃗ = 7 ∆A Quiz: 4 15 ¨ The angular momentum of a body about a particular axis as a function of time is shown in the graph. What is the net external torque acting on the body along this axis at � = 2 s in N·m? Quiz: 24-4 answer 16 ¨ 5 N·m ¨ The slope of this graph represent the net external torque. GH ;C DEF 7 ∆B I ¨ ∑ � = = = 5 N F m 7 ∆A J K Example: 2 17 ¨ A windmill has an initial angular 2 momentum of �O = 8500 kg·m /s. The wind picks up, and ∆� = 5.66 s later, the windmill’s angular momentum is 2 �P = 9700 kg·m /s. What was the torque acting on the windmill, assuming it was constant during this time? Demo: 1 18 ¨ Satellite De-Rotator ¤ A large disc has two smaller discs held tight to its outer edge by retractable lever arms. ¤ As the small discs fly free, they are forced by constraints on the cables to achieve the same angular velocity as the large disc. ¤ The cables on the small disk supply torque on the large disk. ¤ Since their moments of inertia are equal, the small discs take up all the angular momentum of the larger disc, stopping it completely..