Lecture 16

1. Lecture 16 Rotational Dynamics

2. Announcements: • Office hours today 1:00 – 3:00

3. Angular Momentum

4. Angular Momentum Consider a particle moving in a circle of radius r, I = mr2 L = Iω = mr2ω = rm(rω) = rmvt = rpt

5. Angular Momentum For more general motion (not necessarily circular), The tangential component of the momentum, times the distance

6. Angular Momentum For an object of constant moment of inertia, consider the rate of change of angular momentum analogous to 2nd Law for Linear Motion

7. As the moment of inertia decreases, the angular speed increases, so the angular momentum does not change. Conservation of Angular Momentum If the net external torque on a system is zero, the angular momentum is conserved.

8. Figure Skater a) the same b) larger because she’s rotating faster c) smaller because her rotational inertia is smaller A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertiaand spins faster so that her angular momentum is conserved. Comparedto her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be:

9. Where does the “extra” energy come from? KErot = I 2 = L (used L = I ). Because L is conserved, larger  means larger KErot. Figure Skater a) the same b) larger because she’s rotating faster c) smaller because her rotational inertia is smaller A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertiaand spins faster so that her angular momentum is conserved. Comparedto her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be:

10. Where does the “extra” energy come from? KErot = I 2 = L2 (used L = I ). Because L is conserved, larger  means larger KErot. As her hands come in, the velocity of her arms is not only tangential... but also radial. So the arms are accelerated inward, and the force required times the Δr does the work to raise the kinetic energy

11. Conservation of Angular Momentum Angular momentum is also conserved in rotational collisions larger I, same total angular momentum, smaller angular velocity

12. Rotational Work A torque acting through an angular displacement does work, just as a force acting through a distance does. Consider a tangential force on a mass in circular motion: τ = r F Work is force times the distance on the arc: s = r Δθ W = s F W = (r Δθ) F = rF Δθ = τ Δθ The work-energy theorem applies as usual.

13. Rotational Work and Power Power is the rate at which work is done, for rotational motion as well as for translational motion. Again, note the analogy to the linear form (for constant force along motion):

14. Dumbbell II a) case (a) b) case (b) c) no difference d) it depends on the rotational inertia of the dumbbell A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater energy ?

15. Dumbbell II a) case (a) b) case (b) c) no difference d) it depends on the rotational inertia of the dumbbell A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater energy ? If the CM velocities are the same, the translational kinetic energies must be the same. Because dumbbell (b) is also rotating, it has rotational kinetic energy in addition.

16. A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m and mass 0.742 kg. If the bucket is allowed to fall, (a) what is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 1.50 s?

17. A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m and mass 0.742 kg. If the bucket is allowed to fall, (a) What is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 1.50 s? (a) Pulley spins as bucket falls (b) (c)

18. The Vector Nature of Rotational Motion The direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives the sign. Right-hand Rule: your fingers should follow the velocity vector around the circle Optional material Section 11.9

19. The Torque Vector Similarly, the right-hand rule gives the direction of the torque vector, which also lies along the assumed axis or rotation Right-hand Rule: point your RtHand fingers along the force, then follow it “around”. Thumb points in direction of torque. Optional material Section 11.9

20. The linear momentum of components related to the vector angular momentum of the system Optional material Section 11.9

21. Applied tangential force related to the torque vector Optional material Section 11.9

22. Applied torque over time related to change in the vector angular momentum. Optional material Section 11.9

23. Spinning Bicycle Wheel You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will: a) remain stationary b) start to spin in the same direction as before flipping c) start to spin in the same direction as after flipping

24. With the spinning wheel: how is L changing? What is the torque (from gravity) around the supporting point? Which direction does it point? Without the spinning wheel: does this make sense? Why does the wheel not fall? Does this violate Newton’s 2nd?

25. Gravity

26. Newton’s Law of Universal Gravitation Newton’s insight: The force accelerating an apple downward is the same force that keeps the Moon in its orbit. Universal Gravitation

27. The gravitational force is always attractive, and points along the line connecting the two masses: The two forces shown are an action-reaction pair. G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth). If an object is being acted upon by several different gravitational forces, the net force on it is the vector sum of the individual forces. This is called the principle of superposition.

28. Gravitational Attraction of Spherical Bodies Gravitational force between a point mass and a sphere*: the force is the same as if all the mass of the sphere were concentrated at its center. a consequence of 1/r2 (inverse square law) *Sphere must be radial symmetric

29. Gravitational Force at the Earth’s Surface The center of the Earth is one Earth radius away, so this is the distance we use: g ...until altitude becomes comparable to the radius of the Earth. Then the decrease in the acceleration of gravity is much larger: The acceleration of gravity decreases slowly with altitude...

30. Astronauts in the space shuttle float because: In the Space Shuttle a) they are so far from Earth that Earth’s gravity doesn’t act any more b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path d) their weight is reduced in space so the force of gravity is much weaker

31. Astronauts in the space shuttle float because: In the Space Shuttle a) they are so far from Earth that Earth’s gravity doesn’t act any more b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path d) their weight is reduced in space so the force of gravity is much weaker Astronauts in the space shuttle float because they are in “free fall” around Earth, just like a satellite or the Moon. Again, it is gravity that provides the centripetal force that keeps them in circular motion. Follow-up: How weak is the value of g at an altitude of 300 km?

32. Satellite Motion: FG and acp • Does not depend on mass of the satellite! • larger radius = smaller velocity • smaller radius = larger velocity Consider a satellite in circular motion*: Gravitational Attraction: Necessary centripetal acceleration: Relationship between FG and acp will be important for many gravitational orbit problems * not all satellite orbits are circular!

33. A geosynchronous satelliteis one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground. These satellites are used for communications and weather forecasting. How high are they? RE = 6378 km ME = 5.87 x 1024 kg

34. The Moon does not crash into Earth because: Averting Disaster a) it’s in Earth’s gravitational field b) the net force on it is zero c) it is beyond the main pull of Earth’s gravity d) it’s being pulled by the Sun as well as by Earth e) none of the above

35. The Moon does not crash into Earth because: Averting Disaster a) it’s in Earth’s gravitational field b) the net force on it is zero c) it is beyond the main pull of Earth’s gravity d) it’s being pulled by the Sun as well as by Earth e) none of the above The Moon does not crash into Earth because of its high speed. If it stopped moving, it would, of course, fall directly into Earth. With its high speed, the Moon would fly off into space if it weren’t for gravity providing the centripetal force. Follow-up: What happens to a satellite orbiting Earth as it slows?

36. Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A? Two Satellites a) 1/8 b) ¼ c) ½ d) it’s the same e) 2

37. Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A? Two Satellites a) 1/8 b) ¼ c) ½ d) it’s the same e) 2 • Using the Law of Gravitation: • we find that the ratio is . Note the 1/R2 factor

38. Gravitational Potential Energy Very close to the Earth’s surface, the gravitational potential increases linearly with altitude: Gravitational potential energy of an object of mass m a distance r from the Earth’s center: (U =0 at r -> infinity) Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a sign.

39. Energy Conservation Total mechanical energy of an object of mass m a distance r from the center of the Earth: This confirms what we already know – as an object approaches the Earth, it moves faster and faster.

40. Escape Speed Escape speed: the initial upward speed a projectile must have in order to escape from the Earth’s gravity from total energy: If initial velocity = ve, then velocity at large distance goes to zero. If initial velocity is larger than ve, then there is non-zero total energy, and the kinetic energy is non-zero when the body has left the potential well

41. Maximum height vs. Launch speed Speed of a projectile as it leaves the Earth, for various launch speeds

42. Black holes The light itself has mass (in the mass/energy relationship of Einstein), or spacetime itself is curved If an object is sufficiently massive and sufficiently small, the escape speed will equal or exceed the speed of light – light itself will not be able to escape the surface. This is a black hole.

43. Gravity and light Light will be bent by any gravitational field; this can be seen when we view a distant galaxy beyond a closer galaxy cluster. This is called gravitational lensing, and many examples have been found.

44. Kepler’s Laws of Orbital Motion Johannes Kepler made detailed studies of the apparent motions of the planets over many years, and was able to formulate three empirical laws 1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse. Elliptical orbits are stable under inverse-square force law. You already know about circular motion... circular motion is just a special case of elliptical motion Only force is central gravitational attraction - but for elliptical orbits this has both radial and tangential components

45. Kepler’s Laws of Orbital Motion 2. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time. r v Δt This is equivalent to conservation of angular momentum

46. Kepler’s Laws of Orbital Motion 3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power. This can be shown to be a consequence of the inverse square form of the gravitational force.

47. Orbital Maneuvers Which stable circular orbit has the higher speed? How does one move from the larger orbit to the smaller orbit?

48. Binary systems If neither body is “infinite” mass, one should consider the center of mass of the orbital motion

49. Guess My Weight If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value

50. Guess My Weight If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value The weight that a scale reads is thenormal forceexerted by the floor (or the scale). At the equator,you are in circular motion,so there must be anet inward forcetoward Earth’s center. This means that thenormal force must be slightly less than mg. So the scale would register something less than your actual weight.