Accounting for Angular Momentum

1. Accounting for Angular Momentum Chapter 21

2. Objectives • Understand the basic fundamentals behind angular momentum • Be able to define measures of rotary motion and measures of rotation

3. Accounting for Angular Momentum • Linear momentum deals with objects moving in a straight line. • Angular momentum deals with objects that rotate or orbit.

4. Angular Motion Consider a person on a spinning carnival ride. Position, velocity, and acceleration are a function of angular, rather than linear measurements. r q1 q2 Time passes

5. Angular Position and Speed • Position can be described by q • Angular displacement, Dq = q2 - q1 • Average angular speedis • Instantaneous angular speed is

6. Angular Acceleration • Average angular accelerationis • Instantaneous angular acceleration is

7. Linear x = xo + vot + ½aot2 dx/dt = v = vo + aot dv/dt = a = ao Angular q = qo + wot + ½aot2 dq/dt = w = wo + aot dw/dt = a = ao Analogy to Linear Motion

8. Pair Exercise #1 • A carnival ride is rotating at 2.0 rad/s. An external torque is applied that slows the ride down at a rate of –0.05 rad/s2. • How long does it take the ride to come to rest? • How many revolutions does it make while coming to rest?

9. Measures of Rotation Frequency, f is the number of cycles (revolutions) per second Unit: Herzt or Hz = 1 cycle/second • n revolutions sweep out q radians. There are 2p radians per cycle (revolution). Therefore • Frequency is given by Angular frequency, ω,is the amount of radians per second.

10. Measures of Rotation Period, T, is the time it takes to complete one cycle (seconds per revolution)

11. Pairs Exercise #2 A carnival ride completes 2 revolutions per second. What is its frequency, period, and angular velocity?

12. More Basic Equations • Speed (magnitude of velocity) is • Using some basic calculus and algebra you can find (Section 20.1.3):

13. Centripetal Forces Applying Newton’s second law: This is the inward force required to keep a mass in a circular orbit. If the force stops being applied, the mass will fly off tangentially.

14. Centrifugal Force From the Inertial Frame The ball is seen to rotate. Centripetal force keep the ball rotating From a co-rotating frame: a rotating object appears to have an outward force acting on it. This fictitious centrifugal force has the same magnitude as the centripetal force, but acts in the opposite direction.

15. The centripetal (not the centrifugal) force can be used to design a centrifuge, which separates materials based upon density differences.

16. Angular momentum (particles) • Angular momentum (L) is a vector quantity • The direction is perpendicular to the plane of the orbit and follows the right hand rule

17. Moment of Inertia The mass moment of inertia is define as: Thus angular momentum can be expressed as: Which is analogous to linear momentum:

18. Angular Momentum for Bodies Finding the angular momentum of a rotating body requires you to integrate over the volume of the body. (see Figure 21.7)

19. Angular Momentum for Bodies • This gets messy for most shapes (even simple ones), so tables with moments of inertia have be developed for standard shapes (see Table 21.2). • From AutoCAD you can use the moment of inertia from Mass Properties times the density of the material. • Once you have the moment of inertia, the angular momentum is found from the formula:

20. Parallel Axis Theorem Because we want properties about an axis other than the center of mass, we must use the Parallel Axis Theorem, which states that Iaxis=Icg+MD2 where Iaxis and ICG are parallel and D is the distance between them. M is the mass of the part.

21. Using the Theorem • With the mass and CG known, and the desired axis known, we can find the correct moment of inertia. • The axis we want is through the point 55, 25 mm. Because the CG’s X and Y are 35.54, 25 mm, the distance between them is 19.46 mm. • So the the correct moment is 275.46 kg•mm2 + 0.401 kg • (19.46 mm)2 = 427.4 kg•mm2

22. Solution

23. Torque • Torque is a twisting force. It is created by applying a force to an object in an attempt to make the object rotate. (see Figure 21.8) • Like momentum, it is a vector quantity.

24. Conservation of Angular Momentum • Angular momentum is a conserved quantity. • Changing angular momentum of a system • Changing mass (see Figure 21.9) • Applying an external torque

25. Torque and Newton • When Torque is applied to an object, it changes the angular momentum of the object in a manner similar to a force applied to a body changes its linear momentum.

26. Pair Exercise #3 • The previous AutoCAD part is to be spun by applying a torque of 0.001N·m to the hole for 10 seconds. • What is its angular velocity (in rpm’s) at the end of the 10-s period?

27. Systems without Net Momentum Input • Many times you do not have unbalanced torque or mass transfer. In this case the UAE simplifies to: Final Amount=Initial Amount • This is how ice skaters spin faster.

28. Pair Exercise #4 • A space satellite has an electric motor in it with a flywheel attached (see Figure 21.11). The motor causes the flywheel to rotate at 10 rpm. If the moment of inertia of the flywheel is 10 kg·m2 and the satellite is 10000 kg·m2, how long must the motor run to twist the satellite by 10 degrees?