7 Circular Motion & Gravitation

1. 7 Circular Motion & Gravitation • Homework: • 17, 24, 29, 33, 44, 59, 67, 74, 92.

2. centripetal acceleration • circular motion: change in velocity due to turning is toward center, i.e. “centripetal” • acceleration = “tangential + centripetal” • a(tangential) = ra. • a(centripetal) = v2/r (r = radius, v = speed)

3. Centripetal Force • required for circular motion • Fc = mac = mv2/r • Example: • 1.5kg moves in r = 2m circle v = 8m/s. • ac = v2/r = 64/2 = 32m/s/s • Fc = mac = (1.5kg)(32m/s/s) = 48N

4. Curved Road • How much horizontal force is required for a 2000kg car to round a corner, radius = 100m, at a speed of 25m/s? • Answer: F = mv2/r = (2000)(25)(25)/(100) = 12,500N • What percent is this force of the weight of the car? • Answer: % = 12,500/19,600 = 64%

5. Mass on Spring 1 • A 1kg mass attached to spring does r = 0.15m circular motion at a speed of 2m/s. What is the tension in the spring? • Answer: T = mv2/r = (1)(2)(2)/(.15) = 26.7N

6. Mass on Spring 2 • A 1kg mass attached to spring does r = 0.15m circular motion with a tension in the spring equal to 9.8N. What is the speed of the mass? • Answer: T = mv2/r, v2 = Tr/m • v = sqrt{(9.8)(0.15)/(1)} = 1.21m/s

7. Radian Angle Meas.

8. angular measurement • degrees (arbitrary numbering system, e.g. some systems use 400) • radians (ratio of distances) • e.g. distance traveled by object is product of angle and radius.

9. r s r Radians s = arc length r = radius s = rq

10. angular conversions Convert 30°to radians: Convert 1.5 radians to degrees:

11. Angular Motion • radian/second (radian/second)/second

12. motion tangent to circle

13. Angular Equations of Motion Valid for constant-a only

14. Newton’s Law of Gravitation

15. Universal Law of Gravity • all matter is weakly attracted • attraction is inverse-square with distance • G = 6.67x10-11 N·m2/kg2 • Example: Two 100kg persons stand 1.0m apart

16. g vs G • G is universal • g ~ Mass and Radius

17. Kepler’s Laws

18. Kepler’s Laws of Orbits • Elliptical orbits • Equal areas in equal times (ang. Mom.) • Square of year ~ cube of radius

19. Elliptical Orbits • One side slowing, one side speeding • Conservation of Mech. Energy • ellipse shape • simulated orbits

20. s = rq v = rw a(tangential) = ra. a(centripetal) = v2/r F(grav) = GMm/r2 Kepler’s Laws, Energy, Angular Momentum Summary

21. Centrifugal Force • The “apparent” force on an object, due to a net force, which is opposite in direction to the net force. • Ex. A moving car makes a sudden turn to the left. You feel forced to the right of the car. • Similarly, if a car accelerates forward, you feel pressed backward into the seat.

22. rotational speeds • rpm = rev/min • frequency “f” = cycles/sec • period “T” = sec/cycle = 1/f • degrees/sec • rad/sec w = 2pf

23. 7-43 • Merry go round: 24 rev in 3.0min. • W-avg: 0.83 rad/s • V = rw = (4m)(0.83rad/s) = 3.3m/s • V = rw = (5m)(0.83rad/s) = 4.2m/s

24. Rolling Motion v = vcm = Rw

25. Example: Rolling A wheel with radius 0.25m is rolling at 18m/s. What is its rotational rate?

26. Example A car wheel angularly accelerates uniformly from 1.5rad/s with rate 3.0rad/s2 for 5.0s. What is the final angular velocity? What angle is subtended during this time?

27. Ex: Changing Units

28. vt at ac vt ac Rotational Motion r

29. Convert 50 rpm into rad/s. • (50rev/min)(6.28rad/rev)(1min/60s) • 5.23rad/s