1. Circular Motion and Gravitation
Chapter 5
Circular Motion and Gravitation

2. Introduction Circular motion Acceleration is not constant
Cannot be reduced to a one-dimensional problem
Examples
Car traveling around a turn
Parts of the motion of a roller coaster
Centrifuge
The Earth orbiting the Sun
Gravitation
Explore gravitational force in more detail
Look at Kepler’s Laws of Motion
Further details about g
Introduction

3. Uniform Circular Motion – Overview
Circular motion is an example of accelerated motion
It can be analyzed in terms of acceleration and forces
A problem-solving strategy can be applied in a manner similar to one- and two-dimensional motion approaches
Section 5.1

4. Uniform Circular Motion
Assume constant speed
The direction of the velocity is continually changing
The vector is always tangent to the circle
Uniform circular motion is circular motion at constant speed
Section 5.1

5. Uniform Circular Motion, cont.
Examine one trip around the track
The distance traveled is the circumference of the track
The period of the motion, T is
r is the radius of the track
v is the speed of the motion
The period does not depend on the location on the object
The speed depends on the radius of the circle of motion and so depends on location
Speeds are fastest at the outside edge
Section 5.1

6. Centripetal Acceleration
Although the speed is constant, the velocity is not constant
Direction of acceleration
Always directed toward the center of the circle
This is called the centripetal acceleration
Centripetal means “center-seeking”
Magnitude of acceleration
Section 5.1

7. Circular Motion and Forces
Newton’s Second Law can be applied to circular motion:
The force must be directed toward the center of the circle
The centripetal force can be supplied by a variety of physical objects or forces
The “circle” does not need to be a complete circle
Section 5.1

8. Centripetal Force Example
The centripetal acceleration is produced by the tension in the string
If the string breaks, the object would move in a direction tangent to the circle at a constant speed
Section 5.1

9. Problem Solving Strategy – Circular Motion
Recognize the principle
If the object moves in a circle, then there is a centripetal force acting on it
Sketch the problem
Show the path the object travels
Identify the circular part of the path
Include the radius of the circle
Show the center of the circle
Selecting a coordinate system that assigns the positive direction toward the center of the circle is often convenient
Section 5.1

10. Problem Solving Strategy, cont.
Identify the principles
Find all the forces acting on the object
A free body diagram is generally useful
Find the components of the forces that are directed toward the center of the circle
Find the components of the forces perpendicular to the center
Apply Newton’s Second Law for both directions
The acceleration directed toward the center of the circle is a centripetal acceleration
Section 5.1

11. Problem Solving Strategy, final
Solve for the quantities of interest
Check your answer
Consider what the answer means
Does the answer make sense
Section 5.1

12. Centripetal Acceleration Example: Car
A car rounding a curve travels in an approximate circle
The radius of this circle is called the radius of curvature
Forces in the y-direction
Gravity and the normal force
Forces in the x-direction
Friction is directed toward the center of the circle
Section 5.1

13. Car Example, cont.
Since friction is the only force acting in the x-direction, it supplies the centripetal force
Solving for the maximum velocity at which the car can safely round the curve gives
Section 5.1

14. Example: Car on Banked Curve
The maximum speed can be increased by banking the curve
Assume no friction between the tires and the road
The car travels in a circle, so the net force is a centripetal force
There are forces due to gravity and the normal force acting on the car
Section 5.1

15. Banked Curve, cont.
There is a horizontal component of the normal force
Letting the horizontal be the x-direction
The speed at which the car will just be able to negotiate the turn without sliding up or down the banked road is
When θ = 0, v = 0 and you cannot turn on a very icy unbanked road without slipping
Section 5.1

16. Examples of Circular Motion
When the motion is uniform, the total acceleration is the centripetal acceleration
Remember, this means that the speed is constant
The motion does not need to be uniform
Then there will be a tangential acceleration included
Many examples can be analyzed by looking at the two components
Section 5.2

17. Non-Uniform Circular Motion
If the speed is also changing, there are two components to the acceleration
One component is tangent to the circle, at
The other component is directed toward the center of the circle, ac
Section 5.2

18. Circular Motion Example: Vertical Circle
The speed of the rock varies with time
At the bottom of the circle:
Tension and gravity are in opposite directions
The tension supports the rock (mg) and supplies the centripetal force
Section 5.2

19. Vertical Circle Example, cont.
At the top of the circle:
Tension and gravity are in the same direction
Pointing toward the center of the circle
Section 5.2

20. Vertical Circle Example, Final
There is a minimum value of v needed to keep the string taut at the top
Let Ttop = 0
If the speed is smaller than this, the string will become slack and circular motion is no longer possible

21. Circular Motion Example: Roller Coaster
The roller coaster’s path is nearly circular at the minimum or maximum points on the track
There is a maximum speed at which the coaster will not leave the top of the track:
If the speed is greater than this, N would have to be negative
This is impossible, so the coaster would leave the track
Section 5.2

22. Circular Motion Example: Artificial Gravity
Circular motion can be used to create “artificial gravity”
The normal force acting on the passengers due to the floor would be
If N = mg it would feel like the passengers are experiencing normal Earth gravity
Section 5.2

23. Circular Motion Example: Centrifuge
A centrifuge is a device used in many laboratories
It can be used to separate particles or molecules
Or remove them
The effective force causes the particle to move to the bottom of the test tube
Similar to artificial gravity
To someone outside of the test tube, the particle appears to spiral
Section 5.2

24. Frames of Reference and the Centrifuge
The stationary observer is in an inertial reference frame
He can apply Newton’s Second Law
His interpretation is correct
There is no actual force acting on the particle
The force FAG is a fictitious force
The observer moving with the particle is in an accelerated frame
He cannot apply Newton’s Second Law
He thinks there is a force acting on the particle
They agree on the particle’s motion
Section 5.2

25. Newton’s Law of Gravitation
In many cases, the orbits of planets and moons are approximately circular
The Law of Gravitation plays a key role in physics
It allows us to calculate and understand the motion of a wide variety of objects
Newton’s application of his law of gravitation to motion of planets and moons was the first time physics was successfully applied to describe the motion of the solar system
Showed that the laws of physics apply to all objects
Had an effect on how people viewed the universe
Section 5.3

26. Newton’s Law of Gravitation, Equation
Law states: There is a gravitational attraction between any two objects. If the objects are point masses m1 and m2, separated by a distance r the magnitude of the force is
Section 5.3

27. Law of Gravitation, cont.
Note that r is the distance between the objects
G is the Universal Gravitational Constant
G = 6.67 x N . m2/ kg2
The gravitational force is always attractive
Every mass attracts every other mass
The gravitational force is symmetric
The magnitude of the gravitational force exerted by mass 1 on mass 2 is equal in magnitude to the force exerted by mass 2 on mass 1
The two forces form an action-reaction pair
Section 5.3

28. Gravitation and the Moon’s Orbit
The Moon follows an approximately circular orbit around the Earth
There is a force required for this motion
Gravity supplies the force
Section 5.3

29. Notes on the Moon’s Motion
We assumed the Moon orbits a “fixed” Earth
It is a good approximation
It ignores the Earth’s motion around the Sun
The Earth and Moon actually both orbit their center of mass
We can think of the Earth as orbiting the Moon
The circle of the Earth’s motion is very small compared to the Moon’s orbit
Section 5.3

30. Gravitation and g
Assuming a spherical Earth, we can consider all the mass of the Earth to be concentrated at its center
The value of r in the Law of Gravitation is just the radius of the Earth
Section 5.3

31. Gravitation and g, cont.
Since the weight of a person is also the gravitational force between the person and the Earth, we can find the value of g:
The value of g is a function only of the Earth’s mass and radius, and the value of G
Section 5.3

32. Gravitation Force From The Earth – Assumptions
We assumed
A spherical Earth
That the gravitational force could be calculated as if all the mass of the Earth was located at its center
Assumptions are true as long as the density of the object is spherically symmetric
The object has a constant density
The object’s density varies with depth as long as the density depends only on the distance from the center
Section 5.3

33. Measuring G
Henry Cavendish measured the force of gravity between two large lead spheres
By an experimental set-up similar to the picture, he was able to determine the value of G
Section 5.3

34. More About Gravity – Newton’s Apple
G is a constant of nature
Gravity is a weak force
Much smaller than typical normal or tension forces
Gravity is considered the weakest of the fundamental forces of nature
Newton showed the motion of celestial bodies and the motion of terrestrial motion are caused by the same force and governed by the same laws of motion
Section 5.3a

35. Johannes Kepler
Kepler studied results of other astronomer’s measurements of portions of the Moon, planets, etc.
He found the motion of the Moon and planets could be described by a series of laws
Now called Kepler’s Laws of Planetary Motion
Kepler’s Laws are mathematical rules inferred from the available information about the motion in the solar system
Kepler could not give a scientific explanation or derivation of his laws
Newton’s Laws of Motion and Gravitation give the explanation
Section 5.4

36. Kepler’s First Law of Planetary Motion
Planets move in elliptical orbits
The Sun is at one focus
This was very different from the previous idea that the planets moved in perfect circles with the Sun at the center
Section 5.4

37. Kepler’s First Law of Planetary Motion, Planetary Orbits
Section 5.4

38. Kepler’s Second Law of Planetary Motion
A line connecting a planet to its sun sweeps out equal areas in equal times as the planet moves around its orbit
If the time required for the planet to sweep out area A1 is equal to the time to sweep out A2, the areas will be equal
The planet’s speed will be slowest when it is farthest from its sun and fastest when it is closest
Section 5.4

39. Kepler’s Third Law of Planetary Motion
The square of the period of an orbit is proportional to the cube of the orbital radius
For simplicity, apply to a circular orbit
Period is T and the gravitational force supplies a centripetal force
Also applies to satellites with MSun replaced by Mplanet
Section 5.4

40. Orbit Examples Low Earth orbit Geosynchronous Orbit
International Space Station, for example
T ~ 90 minutes
r ~ 6.66 x 106 m
Geosynchronous Orbit
T = 1 day
Always above the same position above the Earth
r = 4.2 x 107 m
Section 5.4

41. Kepler’s Laws and Orbits, Summary
Kepler’s Three Laws of planetary motion apply to all types of gravitationally produced orbital motion
Different motions result from different ways of initially setting objects into motion
For example, launching a satellite to the east into an equatorial orbit takes advantage of the speed of the Earth’s rotation
Section 5.4

42. Tides Tides are the fluctuations of the level of the Earth’s oceans
Tides are due to the Moon’s gravitational force on the oceans
Also exerts a force on the solid Earth
The bulge in the ocean is due to the slightly greater force acting on the side of the Earth facing the Moon
Due to the dependence of gravity on distance
The bulge is a high tide
Section 5.5

43. Tides, cont. One high tide occurs when the Moon is directly overhead
The acceleration of the water closest to the Moon is more than that of the rest of the Earth
Twelve hours later the acceleration of the solid Earth is more than the ocean water on the farther side
There are generally two high tides per day, 12 hours apart
Section 5.5

44. Tides, final The Sun also affects tides
Its effects are smaller than the Moon’s
When the Sun and the Moon are aligned and on the same of the Earth, the tide is higher than when produced by the Moon alone
Section 5.5

45. Inverse Square Law
Newton’s Law of Gravitation is an example of what is called in mathematics an inverse square law
A number of other forces in nature are also inverse square laws
Force between two electric charges, for example
How can you explain why so many forces follow this pattern?
Section 5.6

46. Field Lines
Imagine that an object possesses gravitational field lines that emanate from it
Also imagine that the number of lines is proportional to the mass
When the lines intersect another object, there is a force on that object directed parallel to the lines
Section 5.6

47. Field Lines, cont.
Since the force lines emanate in three-dimensional space, the number of lines that intercept the second object falls off with distance as 1/r2
The result implies that gravity follows an inverse square law because we live in three-dimensional space
It also means we should expect other forces described by the field line picture to have the same inverse square dependence
Section 5.6

48. Field Lines, final No one has devised a way to see the line
You can observe the resulting force that the lines are presumed to cause
Gravitational force is also an example of “action at a distance”
Newton’s Law of Gravitation tells us that action at a distance does occur
The Law doesn’t tell us how it occurs
The field line picture was invented to answer this “how” question
Section 5.6

49. Mass
The mass in the gravitational force is sometimes called gravitational mass
The mass in Newton’s Second Law of Motion is called inertial mass
The gravitational mass of an object is exactly equal to its inertial mass
Needed the theories of relativity to explain why they were equal
Also tells us that a photon of light will be accelerated by gravity
Even though it is a massless particle
Section 5.6