Physics 350

1. Physics 350 Chapter 5 Circular Motion and the Law of Gravity

2. Centripetal Acceleration Consider a car moving in a circle with constant velocity Even though the car is moving with constant speed, it has an acceleration The centripetal acceleration is due to the change in the direction of the velocity

3. Centripetal Acceleration Using vectors, rearrange to determine the change in velocity (direction) The vector change is directed towards the center of motion

4. Pictorial “Derivation” of Centripetal Acceleration a a a a a In uniform circular motion the acceleration is constant, directed towards the center. The velocity has constant magnitude, and is tangent to the path. a = Dv/Dt v2 Top view: v1 a = v2/r (r is radius of curve)

5. Centripetal Acceleration The magnitude of the centripetal acceleration is given by This direction is toward the center of the circle

6. Centripetal Acceleration An object can have a centripetal acceleration only if some external force acts on it. In the case in the figure, the force is the tension in the string

7. Centripetal Acceleration For the car moving on a flat circular track, the force is the friction between the car and the track.

8. Centripetal Acceleration • Forces that act inward are considered to be centripetal forces • Examples: • Tension in example above • Gravity on a satellite orbiting the Earth • Force of friction

9. Centripetal Acceleration • Applying Newton’s Second Law along the radial direction we can determine the net centripetal force Fc Fc = mac = m (v2 / r)

10. Centripetal Acceleration If the centripetal force were removed, the object would leave its circular path and move in a straight line tangent to the circle Merry go round

11. Curves, Centrifugal, Centripetal Forces Going around a curve smushes you against window Understand this as inertia: you want to go straight your body wants to keep going straight but the car is accelerating towards the center of the curve Car acceleration is v2/r  you thinkyou’re being accelerated by v2/r relative to the car

12. Centripetal, Centrifugal Forces, continued The car is accelerated toward the center of the curve by a centripetal (center seeking) force In your reference frame of the car, you experience a “fake”, or fictitious centrifugal “force” Not a real force, just inertia relative to car’s acceleration Centripetal Force on car velocity of car (and the way you’d rather go)

13. Centripetal Forces • Fictitious Force - Centrifugal • Driving in a car around a curve feels like you are applying a force to the car outward • Not really a force, the force one feels is the car applying a force on you from the frictional force it applies to the road • Inertia keeps our bodies wanting to move forward, the car applies a force to push it inwards

14. Rotating Drum Ride Vertical drum rotates, you’re pressed against wall Friction force against wall matches gravity Seem to stick to wall, feel very heavy Real Forces: Perceived Forces: Friction; up Centripetal; inwards Gravity (weight); down Centrifugal; outwards Gravity (weight); down Perceived weight; down and out The forces real and perceived

15. Centripetal Acceleration • Gravitron • Accelerating upwards? • Climbing car

16. Works in vertical direction too… Roller coaster loops: Loop accelerates you downward (at top) with acceleration greater than gravity You are “pulled” into the floor, train stays on track it’s actually the train being pulled into you!

17. Vertical Circular Motion Consider the forces acting on a motorcycle performing a loop-to-loop: Can you think of other objects that undergo similar motions?

18. Old-Fashioned Swings The angle of the ropes tells us where the forces are: Ropes and gravity pull on swingers If no vertical motions (level swing), vertical forces cancel Only thing left is horizontal component pointing toward center: centripetal force Centripetal force is just mv2/r (F = ma; a = v2/r) swing ropes: what you feel from your seat gravity (mg) resultant: centripetal

19. What about our circular motions on Earth? Earth revolves on its axis once per day Earth moves in (roughly) a circle about the sun What are the accelerations produced by these motions, and why don’t we feel them?

20. Earth Rotation Velocity at equator: 2r / (86,400 sec) = 463 m/s v2/r = 0.034 m/s2 ~300 times weaker than gravity, which is 9.8 m/s2 Makes you feel lighter by 0.3% than if not rotating No rotation at north pole  no reduction in g If you weigh 150 pounds at north pole, you’ll weigh 149.5 pounds at the equator actually, effect is even more pronounced than this (by another half-pound) owing to stronger gravity at pole: earth’s oblate shape is the reason for this

21. T W ConcepTest 5.1Tetherball 1) toward the top of the pole 2) toward the ground 3) along the horizontal component of the tension force 4) along the vertical component of the tension force 5) tangential to the circle In the game of tetherball, the struck ball whirls around a pole. In what direction does the net force on the ball point?

22. T W T W ConcepTest 5.1Tetherball 1) toward the top of the pole 2) toward the ground 3) along the horizontal component of the tension force 4) along the vertical component of the tension force 5) tangential to the circle In the game of tetherball, the struck ball whirls around a pole. In what direction does the net force on the ball point? The vertical component of the tension balances the weight. The horizontal component of tension provides the centripetal force that points toward the center of the circle.

23. Centripetal Acceleration • The tangential component of the acceleration is due to changing speed • The centripetal component of the acceleration is due to changing direction • Total acceleration can be found from these components

24. Centripetal Acceleration • Example • A 1,000kg car rounds a curve on a flat road of radius 50.0m at a speed of 50.0km/hr (14.0m/s). Will the car make the turn if: • a) the pavement is dry and the coefficient of static friction is 0.800? • b) the pavement is icy and the coefficient of static friction is 0.200? • (Note: use max static friction here for the extreme case of the tires almost slipping.)

25. Centripetal Acceleration • Example: • An engineer wishes to design a curved exit ramp for a toll road in such a way that a car will not have to rely on friction to round the curve without skidding. He does so by banking the road in such a way that the force causing the centripetal acceleration will be supplied by the component of the normal force toward the center of the circular path. • a) Show that curve must be banked at tan θ= v2/rg. • b) Find the angle at which the curve needs to be banked for a 50m radius and a speed of 13.4 m/s.

26. Planetary Motion and Newtonian Gravitation

27. Kepler’s Laws • All planets move in elliptical orbits with the Sun at one of the focal points. • A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals. • The square of the orbital period of any planet is proportional to the cube of the average distance from the planet to the Sun.

28. Kepler’s Laws • Kepler’s First Law • “All planets move in elliptical orbits with the Sun at one of the focal points.” • Any object bound to another by an inverse square law will move in an elliptical path • Second focus is empty

29. Kepler’s Laws • Kepler’s Second Law • “A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals.” • Objects near the Sun will need to cover more distance per time • Faster velocity

30. Kepler’s Laws • Kepler’s Third Law • “The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet” • Orbital Period = time it takes a planet to make one full orbit around the sun • For orbit around the Sun, T2/r3 =K = KS = 2.97x10-19 s2/m3 • K is independent of the mass of the planet • Therefore, all planets should have the same K

31. Kepler’s Laws • Kepler’s Third Law • They do have the same K!

32. Kepler’s Laws • Planetary Data relative to Earth

33. Newtonian Mechanics • Kepler’s Laws described the kinematics of the motion of the planets but didn’t answer why the planets move the way they do

34. The Universal Law of Gravity • Any two bodies are attracting each other through gravitation, with a force proportional to the product of their masses and inversely proportional to the square of their distance: m1m2 F = G r2 (G is the Universal constant of gravity.)

35. Newtonian Gravitation • Newton’s Universal Law of Gravitation Fg = G ———— where G is the gravitational constant, G = 6.673 x 10-11 m3 kg-1 s-2 • Example of the inverse-square law m1m2 r2

36. Newtonian Gravitation • Applying Newton’s third law to two masses: • Action-Reaction Pair F21 = -F12 “Every pair of particles exerts on one another a mutual gravitational force of attraction.”

37. Newtonian Gravitation • The gravitational force exerted by a uniform sphere on a particle outside the sphere is the same as the force exerted if the entire mass of the sphere were concentrated at its center • This is called Gauss’s Law. • Applies to electric fields also

38. m1 m2 r12 F12 1 kg a liter of soda 1 kg sandwich 1 meter 6.67x10-11 N. 100 kg a person 100 kg another person 1 meter 6.67x10-7 N. 106 kg a ship 106 kg another ship 100 meters 0.67 N. still hard to detect 9.1x10-31 kg electron 1.7x10-27 kg proton 5x10-11 meter orbit radius 4x10-47 N. Why was the Law of Gravitation not obvious (except to Newton).How big are gravitational forces between ordinary objects? 1 Newton is about the force needed to support 100 grams of mass on the Earth About the weight of a small apple • Conclusion: • G is very small, so…need huge masses to get perceptible forces Does gravitation play a role in atomic physics & chemistry?

39. Centripetal acceleration Newton’s second law (a pretty good approximation for all the planets because the eccentricities are much less than 1.) Circular Orbits (velocity) (acceleration) There is a subtle approximation here: we are approximating the center of mass position by the position of the sun. This is a good approximation.

40. Circular Orbits The planetary mass m cancels out. The speed is then Period of revolution Time = distance / speed i.e., Period = circumference / speed  Kepler’s third law: T 2 r 3

41. Generalization to elliptical orbits (and the true center of mass!) where a is the semi-major axis of the ellipse The calculation of elliptical orbits is difficult mathematics.

42. Finding the Value of G • Henry Cavendish first measured G directly (1798) • Two masses m are fixed at the ends of a light horizontal rod (torsion pendulum) • Two large masses M were placed near the small ones • The angle of rotation was measured • Results were fitted into Newton’s Law G=6.67x10-11 N.m2/kg2 • G versus g: • G is the universal gravitational constant, the same everywhere • g = ag is the acceleration due to gravity. It varies by location. • g = 9.80 m/s2 at the surface of the Earth

43. Superposition:The net force on a point mass when there are many others nearby is the vector sum of the forces taken one pair at a time m2 m3 m1 m4 m2 = m3 m4 = m5 Example: m4 m1 m2 m3 m5 All gravitational effects are between pairs of masses. No known effects depend directly on 3 or more masses.

44. Newtonian Gravitation • Acceleration due to gravity • Determined experimentally • g value varies with altitude ag = GME / r2

45. Gravitational “field” transmits the force • The field g1(r) is the gravitational force per unit mass created by mass m1, present at all points whether or not there is a test mass m2 located there • The gravitational field vectors point in the direction of the acceleration a particle would experience if placed in the field at each point. Field lines help to visualize strength and direction. • close together  strong field, • direction  force on test mass • A piece of mass m1 placed somewhere creates a “gravitational field” that has values described by some function g1(r) everywhere in space. • Another piece of mass m2 feels a force proportional to g1(r) and in the same direction, also proportional to m2. Concepts for g-fields: • No contact needed: “action at a distance”. • Acceleration Field created by gravitational mass transmits the force as a distortion of space that another (inertial) mass responds to.

46. Mass of the Earth We can use the equation: g = G*Mearth/Rearth2 to solve for Mearth since we know • g = 9.8 m/s2 (from our lab experiment), • G = 6.67 x 10-11 Nt-m2/kg2 (from precise gravity force experiments), and • Rearth = 6,400 km (since we know the circumference of the earth = 25,000 miles).

47. Mass of the Earth g = G*Mearth/Rearth2 or Mearth = g*Rearth2/G = 9.8 m/s2 * (6.4 x 106 m)2 / 6.67 x 10-11 Nt-m2/kg2 = 6.0 x 1024 kg. Actual value = 5.9742 x 1024kg This value is certainly large as we expect the mass of the earth to be large.

48. ConcepTestFly Me Away You weigh yourself on a scale inside an airplane that is flying with constant speed at an altitude of 20,000 feet. How does your measured weight in the airplane compare with your weight as measured on the surface of the Earth? 1) greater than 2) less than 3) same

49. ConcepTestFly Me Away You weigh yourself on a scale inside an airplane that is flying with constant speed at an altitude of 20,000 feet. How does your measured weight in the airplane compare with your weight as measured on the surface of the Earth? 1) greater than 2) less than 3) same At a high altitude, you are farther away from the center of Earth. Therefore, the gravitational force in the airplane will be less than the force that you would experience on the surface of the Earth.

50. Kepler’s Laws • Example: