Physics I 95.141 LECTURE 10 3/3/10

1. Physics I95.141LECTURE 103/3/10

2. Exam Prep Problem (Conical Pendulum) • A small mass (m) suspended on a cord (l=1m) revolves in a circle of radius r. (θ=30º) • A) (10pts) Draw a free body diagram for the mass, labeling your coordinate system • B) (5pts) What is the acceleration of the ball, and in what direction? • C) (5pts) What is the tangential velocity of the ball? • D) (5pts) (5pts) If you want the frequency of oscillation to double, what could you change the cord length to, assuming a constant angle?

3. Exam Prep Problem (Conical Pendulum) • A small mass (m) suspended on a cord (l=1m) revolves in a circle of radius r. (θ=30º) • A) (10pts) Draw a free body diagram for the mass, labeling your coordinate system

4. Exam Prep Problem (Conical Pendulum) • A small mass (m) suspended on a cord (l=1m) revolves in a circle of radius r. (θ=30º) • B) (5pts) What is the acceleration of the ball, and in what direction?

5. Exam Prep Problem (Conical Pendulum) • A small mass (m) suspended on a cord (l=1m) revolves in a circle of radius r. (θ=30º) • C) (5pts) What is the tangential velocity of the ball?

6. Exam Prep Problem (Conical Pendulum) • A small mass (m=2kg) suspended on a cord revolves in a circle of radius r=0.5m. (θ=30º) • D) (5pts) If you want the frequency of oscillation to double, what could you change the cord length to, assuming a constant angle?

7. Banked Curves • Can a car make a turn on a banked frictionless surface without skidding? For speed v, radius R, what angle is required? • Coordinate system!!

8. Example Problem • A car goes around an unbanked curve (R=100m) at a speed of 50m/s. The concrete/tire interface has a coefficient of static friction of 1. Can the car make this turn?

9. Example Problem • A car goes around an banked curve (R=100m) at a speed of 50m/s. Ignoring friction, what angle should the curve be banked at to allow the car to make the curve?

10. Outline • Newton’s Law of Universal Gravitation • Weightlessness • Kepler’s Laws • What do we know? • Units • Kinematic equations • Freely falling objects • Vectors • Kinematics + Vectors = Vector Kinematics • Relative motion • Projectile motion • Uniform circular motion • Newton’s Laws • Force of Gravity/Normal Force • Free Body Diagrams • Problem solving • Uniform Circular Motion

11. Newton’s Law of Universal Gravitation • We know that falling objects accelerate. • We also know that if an object accelerates, there must be a force acting on it. • The Force that accelerates falling bodies is gravity. • But what exerts this force? • Since all falling objects fall towards the center of the Earth, Newton suggested that it is the Earth itself which is exerting this Force.

12. Newton’s Law of Universal Gravitation • What form does this Force take? • 1) Dependence on distance • Newton knew that the moon orbited the Earth. • We know that a circular motion requires an inward radial acceleration

13. Newton’s Law of Universal Gravitation • So if the Force causing the moon to orbit the Earth, is the same force which causes object to accelerate at the surface of the Earth, then this Force goes as the inverse square of the distance from the center of the Earth

14. Newton’s Law of Universal Gravitation • The other thing we should note, is that the Force due to gravity produces the same acceleration for ALL OBJECTS, regardless of mass. So this is a Force which must scale with the mass of the object. • Symmetry also suggests that this Force must depend on the mass of the Earth, or the second body.

15. Newton’s Law of Universal Gravitation • Finally, Newton argues that if this is the Force causing the moon to orbit the Earth, perhaps it is also the Force causing the planets to Orbit the sun. In fact, perhaps every mass exerts a gravitational force on every other mass in the universe. • And we can write this force as

16. Direction of Gravitational Force • Force is a vector, and therefore has a magnitude and direction. • Direction is along line connecting two masses.

17. Example Problem • Imagine 3 Blocks, of equal mass, placed at three corners of a square. Draw the gravitational Force vectors acting on each block. 1 3 2

18. Gravitational Attraction Between Two People • Tom Cruise (160lbs) and Katie Holmes (118lbs) are dancing (about 0.5 m apart). What is attractive Force of Gravity between them?

19. Gravity at Earth’s Surface • If this law is correct, what should we get for Fg at the Earth’s surface?

20. Satellites • Imagine I throw a ball with some horizontal velocity vo. • In previous chapters, we studied projectile motion, which tells us the ball will accelerate towards earth and eventually fall to Earth. • But this is an approximation…the Earth is not flat, and the Force of gravity is not “downward”, but towards the center of the Earth.

21. Satellites • In order for an object to travel with uniform circular motion, a radial Force is required. • What speed would the ball need to have to travel in a circular path at the surface of the Earth?

22. Satellites • The uniform circular motion due to gravitation is known as an “orbit”, and an orbiting object is often referred to as a “satellite”. • We can calculate the speed of a satellite for a given orbital radius

23. Geosynchronous Orbit • A geosynchronous orbit, is one that orbits at the same speed the Earth rotates, so that the satellite stays at the same position with respect to Earth as it orbits. How can we calculate the height of a satellite of mass M in geosynchronous orbit?

24. Weightlessness • Remember, we define weight as the magnitude of the Force of gravity acting on an object. • At the surface of Earth this is mg. • But we measure weight, by measuring the Force a mass exerts on a scale. • Imagine we are weighing a mass in an elevator. • If the elevator is at rest, or moving at a constant velocity, what does the scale read?

25. Weightlessness • Now, if the elevator is accelerating upwards with a=g/2. • What is the mass’ apparent weight?

26. Weightlessness • Now, if the elevator is in freefall (a=-g). • What is the mass’ apparent weight?

27. Weightlessness • The “weightlessness” one experiences in orbit is exactly the same one would feel in a freely falling elevator. • Remember, the Force causing a satellite to orbit is the Force of Gravity….in essence, the satellite is freely falling.

28. Kepler’s Laws • Kepler’s laws of planetary motion • Empirical (Experimental) • Kepler’s 1st Law: The path of each planet around the sun is an ellipse with the sun at one focus a  semi-major axis b  semi-minor axis e  eccentricity e  es/a es a b

29. Kepler’s Laws D t D t • Kepler’s 2nd Law: Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal periods of time

30. Kepler’s Laws • Kepler’s 3rd Law: The ratio of the squares of the periods of any two planets revolving around the sun is equal to the ratio of the cubes of their semi-major axes. s2 s1

31. Newton’s Synthesis • Newton was able to derive Kepler’s experimental laws from his Universal Law of Gravity. • Perturbations

32. Kepler’s 3rd Law: Circular Orbit • Earth’s orbit around sun has e=0.017, almost circular. Can we prove Kepler’s third law for circular orbits?

33. Example Problem • If we know the Earth’s distance from the sun, can we determine the sun’s mass from Kepler’s laws?