Chapter 11A Angular Motion. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Transcription

1 Chaper 11A Angular Moion A PowerPoin Presenaion by Paul E. Tippens, Proessor o Physics Souhern Polyechnic Sae Universiy 007

2 WIND TUBINES such as hese can generae signiican energy in a way ha is environmenally riendly and renewable. The conceps o roaional acceleraion, angular velociy, angular displacemen, roaional ineria, and oher opics discussed in his chaper are useul in describing he operaion o wind urbines.

3 Objecives: Aer compleing his module, you should be able o: Deine and apply conceps o angular displacemen, velociy, and acceleraion. Draw analogies relaing roaional-moion parameers (,(, ) ) o linear ( (x, x, v, a) a and solve roaional problems. Wrie and apply relaionships beween linear and angular parameers.

4 Objecives: (Coninued) Deine momen o ineria and apply i or several regular objecs in roaion. Apply he ollowing conceps o roaion: 1. oaional work, energy, and power. oaional kineic energy and momenum 3. Conservaion o angular momenum

5 oaional Displacemen, Consider a disk ha roaes rom A o B: B A Angular displacemen : Measured in revoluions, degrees, or radians. 1 rev = = rad The bes measure or roaion o rigid bodies is is he radian.

6 Deiniion o he adian One radian is he angle subended a he cener o a circle by an arc lengh s equal o he radius o he circle. s s 1 rad = =

7 Example 1: A rope is wrapped many imes around a drum o radius 50 cm.. How many revoluions o he drum are required o raise a bucke o a heigh o 0 m? m s 0 m 0.50 m Now, 1 rev = rad = 40 rad 1 rev rad 40 rad h = 0 m = 6.37 rev

8 Example : A bicycle ire has a radius o 5 cm.. I he wheel makes 400 rev,, how ar will he bike have raveled? 400 rev = 513 rad rad 1 rev s = = 513 rad (0.5 m) s = 68 m

9 Angular Velociy Angular velociy, is he rae o change in angular displacemen. (radians per second.) Angular velociy in rad/s. Angular velociy can also be given as he requency o revoluion, (rev/s or rpm): Angular requency (rev/s).

10 Example 3: A rope is wrapped many imes around a drum o radius 0 cm.. Wha is he angular velociy o he drum i i lis he bucke o 10 m in 5 s? s s 10 m 0.0 m = 50 rad 50 rad 5 s = 10.0 rad/s h = 10 m

11 Example 4: In he previous example, wha is he requency o revoluion or he drum? ecall ha = 10.0 rad/s. or 10.0 rad/s 1.59 rev/s rad/rev Or, since 60 s = 1 min: rev 60 s rev s 1 min min = 95.5 rpm h = 10 m

12 Angular Acceleraion Angular acceleraion is he rae o change in angular velociy. (adians per sec per sec.) Angular acceleraion (rad/s ) The angular acceleraion can also be ound rom he change in requency, as ollows: ( ) Since

13 Example 5: The block is lied rom res unil he angular velociy o he drum is 16 rad/s aer a ime o 4 s. Wha is he average angular acceleraion? 0 o or 16 rad/s rad s s h = 0 m = 4.00 rad/s

14 Angular and Linear Speed From he deiniion o angular displacemen: s = Linear vs. angular displacemen v s v = Linear speed = angular speed x radius

15 Angular and Linear Acceleraion: From he velociy relaionship we have: v = Linear vs. angular velociy v v v v a = Linear accel. = angular accel. x radius

16 Examples: Consider la roaing disk: = 0; = 0 rad/s = 4 s Wha is inal linear speed a poins A and B? 1 A 1 = 0 cm = 40 cm B v A = A 1 = (0 rad/s)(0. m); v A = 4 m/s v A = B 1 = (0 rad/s)(0.4 m); v B = 8 m/s

17 Acceleraion Example Consider la roaing disk: 1 A = 0; = 0 rad/s = 4 s Wha is he average angular and linear acceleraion a B? 1 = 0 cm = 40 cm B 0 0 rad/s 4 s = 5.00 rad/s a = = (5 rad/s )(0.4 m) a=.00 m/s

18 Angular vs. Linear Parameers ecall he deiniion o linear acceleraion a rom kinemaics. a v v 0 Bu, a = and v =,, so ha we may wrie: v a v 0 becomes Angular acceleraion is he ime rae o change in angular velociy. 0 0

19 A Comparison: Linear vs. Angular v0 v s v v v a o s v 1 a 0 1 s v a as v v 0 0 o

20 Linear Example: A car raveling iniially a 0 m/s comes o a sop in a disance o 100 m. Wha was he acceleraion? Selec Equaion: as v v0 100 m v o = 0 m/s v = 0 m/s 0 - v o a = = s -(0 m/s) (100 m) a = -.00 m/s

21 Angular analogy: A disk ( = 50 cm), roaing a 600 rev/min comes o a sop aer making 50 rev. Wha is he acceleraion? Selec Equaion: rev rad1 min rad/s min 1 rev 60 s 0 o = 600 rpm = 0 rpm = 50 rev 50 rev = 314 rad 0 - o = = -(6.8 rad/s) (314 rad) = -6.9 m/s

22 Problem Solving Sraegy: Draw and label skech o problem. Indicae + direcion o roaion. Lis givens and sae wha is o be ound. Given:,, (,,,,) Find:, Selec equaion conaining one and no he oher o he unknown quaniies, and solve or he unknown.

23 Example 6: A drum is roaing clockwise iniially a 100 rpm and undergoes a consan counerclockwise acceleraion o 3 rad/s or s. Wha is he angular displacemen? Given: o = -100 rpm; = s = + rad/s rev 1 min rad rad/s min 60 s 1 rev ( 10.5)() (3)() o 1 1 = -0.9 rad + 6 rad = rad Ne displacemen is clockwise (-)

24 Summary o Formulas or oaion v0 v s v v v a o s v 1 a 0 1 s v a as v v 0 0 o

25 CONCLUSION: Chaper 11A Angular Moion