Rotation Angular Momentum Conservation of Angular Momentum

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1 Rotation Angular Momentum Conservation of Angular Momentum Lana Sheridan De Anza College Nov 29, 2017

2 Last time Definition of angular momentum relation to Newton s 2nd law angular impulse angular momentum of rigid objects

3 Overview Finish angular momentum of rigid objects example conservation of angular momentum applying conservation of momentum

4 Example Rigid Object Angular Momentum Part 2: Find an expression for the magnitude of the angular acceleration of the system when the seesaw makes an angle θ with the horizontal. y s t O u x m d g S Figure 11.9 (Example 11.6) A father and daughter demonstrate angular momentum on a seesaw. S m f g of rotation (Thein seesaw Figure is11.9. modeled The rotating as a rigid system rodhas of mass angular M.) Last lecture we found: father and daughter and model them both as particles. The rt of the example is categorized I = 1 ( ) l 2 ( ) l 2 as a substitution problem. 12 Ml2 + m f + m d 2 2 the moments of inertia of the three components: the seesaw

5 Example Rigid Object Angular Momentum Magnitude of the angular acceleration when the seesaw is at an angle θ?

6 Example Rigid Object Angular Momentum Magnitude of the angular acceleration when the seesaw is at an angle θ? Use τ net = Iα

7 Example Rigid Object Angular Momentum Magnitude of the angular acceleration when the seesaw is at an angle θ? Use τ net = Iα τ net = i r i F i = m f g l 2 sin(90 θ) m dg l sin(90 + θ) 2 = g l 2 cos θ(m f m d ) α = τ I = 2g cos θ(m f m d ) l( 1 3 M + m f + m d )

8 Conservation of Angular Momentum 11.4 Analysis Model: Isolated System (An el For an isolated system, ie. a system with no external torques, total angular Isolated momentum System is conserved. (Angular Momentum) m rotates about is no net external ystem, there is no ngular momenm: t 5 0 (11.18) w of conservamomentum to a oment of inertia System boundary Angular momentum The angular momentum of the isolated system is constant. I i v i 5 I f v f 5 constant (11.19) 1 Figures from Serway & Jewett. Examples: star collapses much higher r proportional t Kepler s third quantum num conserve angu trino must be angular mome

9 Conservation of Angular Momentum For zero net external torque: ( ) d L i = 0 dt i

10 Conservation of Angular Momentum For zero net external torque: ( ) d L i = 0 dt i This corresponds to a rotational symmetry in the equations of motion.

11 Conservation of Angular Momentum For zero net external torque: ( ) d L i = 0 dt i This corresponds to a rotational symmetry in the equations of motion. (And now isolated means no external torques act on the system.)

12 Conservation of Angular Momentum Suppose we have a collection of massive particles, which together have some moment of inertia I i at time t = t i, and an angular velocity ω i. Then L i = I i ω i

13 Conservation of Angular Momentum Suppose we have a collection of massive particles, which together have some moment of inertia I i at time t = t i, and an angular velocity ω i. Then L i = I i ω i If the system is isolated, L i = L f.

14 Conservation of Angular Momentum Suppose we have a collection of massive particles, which together have some moment of inertia I i at time t = t i, and an angular velocity ω i. Then L i = I i ω i If the system is isolated, L i = L f. Then, even if the arrangement of the particles is changing (I(t)): I i ω i = I f ω f

15 Example Conservation of Angular Momentum A star rotates with a period of 30 days about an axis through its center. The period is the time interval required for a point on the star s equator to make one complete revolution around the axis of rotation. After the star undergoes a supernova explosion, the stellar core, which had a radius of km, collapses into a neutron star of radius 3.0 km. Determine the period of rotation of the neutron star. 0 Serway & Jewett, page 347.

16 Example Conservation of Angular Momentum Period of rotation of neutron star? Use: I i ω i = I f ω f and ω = 2π T

17 Example Conservation of Angular Momentum Period of rotation of neutron star? Use: I i ω i = I f ω f and ω = 2π T I i = 2 5 M ( m) 2 ; I f = 2 5 M ( m) 2

18 Example Conservation of Angular Momentum Period of rotation of neutron star? Use: I i ω i = I f ω f and ω = 2π T I i = 2 5 M ( m) 2 ; I f = 2 5 M ( m) 2 I i 2π T i = I f 2π T f T f = I f I i T i = ( m) 2 ( ( s) m) 2 = 0.23 s

19 n- hemical energy the angular does the speed woman s about body the center convert of into mass? n echanical energy of the woman turntable system as 37. A wooden block of mass M resting on a frict he d Angular woman sets herself Momentum and the v Conservation Inel. Collision S turntable into motion? S horizontal surface is attached to a rigid rod of l /s student sits on a freely rotating stool holding two and of negligible mass (Fig. P11.37). The rod is le umbbells, each of mass m kg (Fig. P11.34). When at the other end. A bullet of mass m traveling gy is arms are extended horizontally (Fig. P11.34a), the to the horizontal surface and perpendicular to umbbells e are 1.00 m from the axis of rotation and the with speed v hits the block and becomes embed tudent m e rotates #37, with page an angular 359speed 2 of rad/s. it. (a) What is the angular momentum of the he moment of inertia at a of the student plus bstool is block system about a vertical axis through the.00 kg m h 2 and is assumed to be constant. The student (b) What fraction of the original kinetic energ ulls the dumbbells inward horizontally Figure P11.36 to a position bullet is converted into internal energy in the to.300 m from the rotation axis (Fig. P11.34b). (a) Find during the collision? he as new 37. angular A wooden speed block of the of student. mass M (b) resting Find the on a frictionless, inetic? energy S horizontal of the rotating surface system is attached before to and a rigid after rod of length, e o pulls the and dumbbells of negligible inward. mass (Fig. P11.37). The rod is pivoted M n at the other end. A bullet of mass m traveling parallel S v e to vthe i horizontal surface vand f perpendicular to the rod m e with speed v hits the block and becomes embedded in s. it. (a) What is the angular momentum of the bullet is block system about a vertical axis through the pivot? nt (b) What fraction of the original kinetic energy of the n bullet is converted into internal energy in the system Figure P11.37 d during the collision? e er a Figure P11.34 M uniform cylindrical turntable of radius m 1.90 m and ass 30.0 kg rotates counterclockwise in a horizontal b S v 38. Review. A thin, uniform, rectangular signboard vertically above the door of a shop. The sign is to a stationary horizontal rod along its top ed mass of the sign is 2.40 kg, and its vertical dim is 50.0 cm. The sign is swinging without frictio is a tempting target for children armed with sno The maximum angular displacement of the

20 Angular Momentum Conservation Inel. Collision (a) Angular momentum about pivot?

21 Angular Momentum Conservation Inel. Collision (a) Angular momentum about pivot? L b,i = m v l = L f (clockwise; downward)

22 Angular Momentum Conservation Inel. Collision (a) Angular momentum about pivot? L b,i = m v l = L f (clockwise; downward) (b) Fraction of initial KE lost as internal energy?

23 Angular Momentum Conservation Inel. Collision (a) Angular momentum about pivot? L b,i = m v l = L f (clockwise; downward) (b) Fraction of initial KE lost as internal energy? K = K f K i = 1 2 Iω2 1 2 mv 2

24 Angular Momentum Conservation Inel. Collision (a) Angular momentum about pivot? L b,i = m v l = L f (clockwise; downward) (b) Fraction of initial KE lost as internal energy? K = K f K i = 1 2 Iω2 1 2 mv 2 I = (M + m)l 2 ; Iω 2 = (M + m)v 2 f Conservation of angular momentum: mvl = (M + m)v f l

25 Angular Momentum Conservation Inel. Collision (a) Angular momentum about pivot? L b,i = m v l = L f (clockwise; downward) (b) Fraction of initial KE lost as internal energy? K = K f K i = 1 2 Iω2 1 2 mv 2 I = (M + m)l 2 ; Iω 2 = (M + m)v 2 f Conservation of angular momentum: mvl = (M + m)v f l K K i = M M + m

26 Summary angular momentum of rigid objects conservation of angular momentum (Uncollected) Homework Play with office furniture Serway & Jewett: Ch 11, onward from page 357. Probs: 31, 33, 37, 39, 41

Rotation Angular Momentum

Rotation Angular Momentum Lana Sheridan De Anza College Nov 28, 2017 Last time rolling motion Overview Definition of angular momentum relation to Newton s 2nd law angular impulse angular momentum of rigid