Angular Momentum: Lecture 24 by Kazumi Tolich, Slides of Acting

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LECTURE 24

ANGULAR MOMENTUM

Instructor: Kazumi Tolich

Lecture 24

¨ Reading chapter 11- 6

¤ Angular momentum n Angular momentum about an axis n Newton’s 2nd^ law for rotational motion 2

Angular momentum in general

¨ Angular momentum of a point particle about an axis is defined by 𝐿 = 𝑟𝑝 sin 𝜃 = 𝑟𝑚𝑣 sin 𝜃 = 𝑝-𝑟 = 𝑝𝑟. ¤ 𝐫⃗ : position vector for the particle from the axis. ¤ 𝐩: linear momentum of the particle: 𝐩 = 𝑚𝐯 ¤ 𝑟. is moment arm, or sometimes called “perpendicular distance.” 4 𝐫⃗ 𝐩 𝐩- 𝜃 Axis 𝑟.

Quiz: 1

¨ 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 B. Only Case B C. Neither D. Both 5 Case A Case B

Sign convention of L

¨ The sign of 𝐿 is determined by the angle to the object.

¤ Positive 𝐿 for increasing 𝜃: an object in (a) or an object moving counterclockwise in (b). ¤ Negative 𝐿 for decreasing 𝜃 : opposite motions from (a) or (b). 7

Example: 1

¨ You observe a particle with a mass 𝑚 = 2.0 kg moving at a constant speed of 𝑣 = 3.5 m/s in a clockwise direction around a circle of radius 𝑟 = 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? 8

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

¨ 𝐿 8 < 𝐿: < 𝐿; ¨ 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: 𝐿 = 𝑝𝑟. ¤ Particle 1: 1 unit ¤ Particle 2: 3 units ¤ Particle 3: 2 units 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. ¨ 𝐿 = 𝑝𝑟. Velocity vectors are shown.

Newton’s 2 nd law for angular motion ¨ 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. 5 𝜏

¨ This is analogous to Newton’s 2 nd law for linear motion using linear momentum, ∑^ 𝐅⃗ ^ =^ ∆𝐩 ∆A 14

Quiz: 24-4 answer

¨ 5 N·∙m ¨ The slope of this graph represent the net external torque. ¨ ∑ 𝜏 ^ =^ ∆B ∆A

;C DEF GH I J K = 5 N F m 16

Example: 2

¨ A windmill has an initial angular momentum of 𝐿O = 8500 kg·∙m 2 /s. The wind picks up, and ∆𝑡 = 5.66 s later, the windmill’s angular momentum is 𝐿P = 9700 kg·∙m 2 /s. What was the torque acting on the windmill, assuming it was constant during this time? 17