1. Circular Motion & Gravity

2. Circular Motion Objects travel in a circle
Rotate about an axis of rotation
Tangential speed (vt) describes the rate at which the object moves around the circle
Direction is tangential to the circular path

3. vt depends upon radius Given the object is rigid, e.g. a CD
Object B must travel a greater distance to keep up with object A
SB > SA
But ΔtB = ΔtA
Therefore, vB > vA

4. Comparison of Translational Motion & Uniform Circular Motion
UCM = motion of an object traveling in a circle at a constant speed, vt
Type of Motion
Translational
Uniform Circular
Displacement
Linear Δx
Circumference
2πr
Time Δt
Period T
Formula
vavg = Δx/Δt
vt = 2πr/T

5. Uniform Circular Motion
Tangential speed vt is constant
Because direction is changing, there is acceleration
Centripetal acceleration

6. Centripetal Acceleration
a = Δv/Δt
When subtracting vectors, reverse the direction of vi
Centripetal acceleration is, therefore, directed toward the center (axis of rotation) when θ is small

7. Centripetal Acceleration
Centripetal means “center seeking” and is always directed toward the center
Due to a change in direction of vt
Phet simulation

8. Tangential Acceleration
Tangential acceleration occurs when there is a change in tangential speed.
For example, if a car is speeding up as it goes around a curve,
It has tangential acceleration and
Centripetal acceleration

9. Centripetal Force
Because Fc acts at right angles to the object’s circular motion, it changes the direction of the objects velocity

10. Centripetal Force Is the cause of centripetal acceleration
It is directed toward the axis of rotation
It is the net force acting on an object in uniform circular motion, i.e. it is the cause of circular motion
Centrifugal force is a misunderstanding of inertia

11. Centripetal Force & Newton’s 2nd Law

12. Centripetal Force
Is just the name of any net force acting on an object in uniform circular motion
Fc could take any form….
It could be frictional force, tension force, gravitational force, etc.

13. Motion of a Car Around a Curve
On a horizontal turn, the centripetal force is friction

14. Circular Motion About a Banked Curve

15. Conical Pendulum

16. Vertical Circular Motion

17. Centifugal Force?
If Fc is insufficient to maintain circular motion, the object will leave it’s circular path due to its own inertia, not because some force is pulling it away from the axis of rotation
Thus, inertia is often mistaken for “centrifugal force”

18. Gravity

19. Gravitational Force Force of attraction between two masses
Attractive only
One of four fundamental forces
Very weak (the weakest)
When one object orbits another, gravitational force is a centripetal force

20. Newton’s Law of Universal Gravitation
Gravitational force is…
directly proportional to the product of the masses of the two bodies
inversely proportional to the square of the distance between the centers of the two masses
If the objects are large (e.g. planets, moons) then the radii would be included in r

21. Gravitational Force Exists Between Any Two Masses

22. Newton’s Cannon http://spaceplace.nasa.gov/en/kids/orbits1.shtml

23. Importance of Gravitational Force
Keeps you from floating away into space
Gravitational force keeps the Moon and planets in orbit
Keeps earth in orbit around sun
Causes ocean tides

24. Black Holes: Extreme Gravity
Extreme density
Escape velocity > speed of light
Detect by effects on surrounding matter

25. Gravitational Field Strength
Increases as distance from mass center decreases
Because gravitational field strength varies, weight varies with location

26. Gravitational Field Strength
Describes the amount of gravitational force per unit mass at any given point
Equals free-fall acceleration

27. Weight Changes with Location
Because gravitational field strength varies, ag varies (acceleration of gravity).
Since w = mag, weight must vary as ag varies
Fg is an example of an inverse square law

28. 7.3 Motion in Space Astronomer Planets orbit… Type of orbit Ptolomey
Earth
Epicycles
Copernicus
Sun
Circular
Kepler
Elliptical

29. Kepler’s Laws of Planetary Motion
1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.
2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times.
3. The Law of Periods: The square of the period of any planet is proportional to the cube of the average distance from the sun,

30. Kepler’s 1st and 2nd Laws
Kepler's Law Simulation

31. Kepler’s 3rd Law Describes Orbital Period

32. Actual and Apparent Weight
A bathroom scale records the normal force of scale acting on your body
Step on the scale … the normal force equals your weight

33. Actual and Apparent Weight
Now try this
Step on the scale and have someone press down on your shoulders
Predict and explain the result
Step on the scale and have someone lift you slightly

34. Actual and Apparent Weight
How does this relate to your experiences in an elevator?
What would the scale read if, in an elevator, it descended with an acceleration of g?

35. Weight and Apparent Weightlessness

36. Torque
a quantity that measures the ability of a force to rotate an object about an axis
is not a force
“rotating ability”
the product of force and “lever arm”
τ = F · d sinθ
Lever arm (d) is distance perpendicular to direction of force to axis of rotation

38. Torque Sign (+) is counterclockwise (-) is clockwise Net Torque and
when 2 or more forces act to rotate the same object, τnet = Στ
τnet = τ1 + τ2 = F1d1 + F2d2

39. Torque Equilibrium
Torque Equilibrium: Στ = 0

40. Torque Equilibrium
The torque due to the boy is equal and opposite to that of the girl.

41. Net Torque

42. Center of Mass (COM) Point mass vs. extended object
The point in a body at which all the mass can be considered to be concentrated when analyzing translational motion
Unless an object rotates about a fixed point, (e.g. a hinge)…
The point about which a mass or system of mass rotates during rotational motion

43. Center of Mass The extended object rotates about the CoM
CoM follows the expected parabolic path

44. Center of Mass
May not lie within the mass or system of masses

45. Simple Machines All machines are combinations of simple machines
Purpose is to change magnitude or direction of an input force
Mechanical Advantage
describes the ratio of output and input forces

46. Ideal vs. Actual Mechanical Advantage
Ideal MA
MA if there were no friction
Actual MA
MA that takes friction into account

47. Machines and Work Machines do not change the amount of work
Machines make work easier

48. Efficiency A measure of how well a machine works
A less efficient machine produces less output per input
A less efficient machine requires more input to get the same output