1. Force and Motion Before Newton
Galileo
Determined that objects fall at the same rate, regardless of their weight
Concluded that motion is a natural state (an object in motion will stay in motion until something stops it)
René Descartes (1596–1650)
Undisturbed motion continues in a straight line
Self-sustaining circular motion therefore not possible
Kepler
Motion of planets due to “paddle” effect of Sun’s rays (“anima motrix” force)
Magnetism accounted for ellipticity of planetary orbits
Robert Hooke (1635–1703)
Showed that a central force could lead to orbital motion
Falling objects and planetary orbits are due to same force

2. Isaac Newton (1642 – 1727)
Kepler’s Laws provide an accurate description of planetary orbits, but do not explain them
Newton provided the full explanation
Newton lived in England and was educated at Trinity College in Cambridge
Shortly after Newton received B.A. degree (1665), a plague epidemic shut down Cambridge
During the next 2 years at his family farm, Newton made major advances in several fields
Invented calculus (needed to solve his new equations)
Fundamental discoveries in optics (later included development of the first reflecting telescope)
Fundamental and universal discoveries in mechanics
Newton published his findings in the Principia (1687)

3. Newton’s Laws of Motion
Newton’s 1st Law: An object remains at rest or continues in motion at constant velocity unless it is acted on by an unbalanced external force
Often called the law of inertia (since inertia describes an object’s resistance to changes in motion)
Essentially the same as the description of Descartes
Led Newton to realize that planets must be attracted to Sun by some force
Velocity is the rate position changes with time
Depends on speed (magnitude) and direction of motion
Acceleration is the rate velocity changes with time
Nonzero when speed and/or direction of motion changes
Force is a push or a pull that affects motion
Can act when objects touch (contact forces) or even when objects are separated by some distance (like gravity)

4. Newton’s Laws of Motion
Momentum (p) is the product of mass (m) and velocity (v):
Mass is an intrinsic property of an object that describes its resistance to acceleration and amount of material in object
Newton’s 2nd Law: When a net force (Fnet) acts on an object, it produces a change in the momentum of the object in the direction in which the force acts
A change in momentum could be a change in mass and/or velocity
Usually, mass remains constant, in which case:
An object accelerates when the sum of all forces acting on it is not zero
Zero acceleration does not necessarily mean that there are no forces acting on an object

5. Newton’s Laws of Motion
Newton’s 3rd Law: When one body exerts a force on a second body, the second body also exerts a force on the first. These forces are equal in strength, but opposite in direction
Sometimes called law of action and reaction
“Every action has an equal and opposite reaction”
Forces always come in pairs (action–reaction pairs)
Action–reaction pair forces always act on different objects
When a small car and large truck collide, both experience the same force and thus the same change in momentum (but the less massive car experiences a larger change in velocity)
Total momentum of interacting objects is conserved (remains the same) when forces are acting between them

6. Newton’s Development of Gravity
A key step in Newton’s understanding of gravity was the calculation of an object’s centripetal acceleration during circular motion
We know an acceleration must be present because object’s velocity keeps changing
Associated with a centripetal acceleration is a centripetal force
Responsible for the circular motion
Familiar example: ball whirling around on a string (tension in string provides centripetal force)

7. Newton’s Development of Gravity
Newton found that a falling apple experiences the same type of force (gravity) as the Moon orbiting the Earth
In Newton’s own words: I thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth, and found them answer pretty nearly.
Newton’s Law of Gravitation: Every 2 particles of matter in the universe attract each other through the gravitational force
Magnitude of the gravitational force:
G = 6.67  10–11 Nm/kg2
M, m are the 2 interacting masses
d = distance between M and m
Direction always indicates attraction
Equal magnitude forces in opposite directions M FG d FG m

8. Newton’s Development of Gravity
Newton also showed (using integral calculus) that the gravitational force between 2 spheres is the same as the force between 2 point masses:
The weight of an object is the gravitational force attracting an object to the Earth:
g = acceleration of gravity = 9.8 m/s (or 32 ft/s2) at Earth’s surface
W and g depend on location (unlike mass)
Weight on Moon < Weight on Earth (but mass is same) M m M FG FG m FG FG = d d
(spherically symmetric masses)
(point masses concentrated at centers)
(R = distance from Earth’s center)

9. Newton’s Proofs of Kepler’s Laws
Newton showed that elliptical orbits are the natural result of gravity having a dependence
Thus proving Kepler’s First Law
Newton showed that any body orbiting another body under the force of gravity moves so that it sweeps out equal areas in equal times
Consequence of conservation of angular momentum (momentum associated with rotation or revolution)
Angular momentum is proportional to the product of orbital distance and transverse velocity
As orbital distance decreases, transverse velocity increases (keeping angular momentum constant)
These calculations proved Kepler’s Second Law

10. Newton’s Proofs of Kepler’s Laws
Newton was able to derive a more general form of Kepler’s Third Law using his laws of motion and the law of gravitation:
P = orbital period
a = average distance between the two bodies
m, M are the masses of the two bodies
If m = mass of planet, and M = mass of Sun, then m + M ≈ M
That’s why Kepler found that P2 / a3 is the same for each planet
These results are extremely useful for determining the masses of astronomical objects
Newton never determined how gravity works at a distance
Took Einstein’s General Theory of Relativity to explain it

11. Orbital Energy and Speed
Another way to look at the changing speed of an orbiting body is through the energy of the body
Energy of an orbiting body has two components: kinetic and gravitational potential energy
Kinetic energy = energy of motion
Gravitational potential energy = energy of position
The larger the distance from the Sun, the larger the gravitational potential energy
The smaller the distance from the Sun, the smaller the gravitational potential energy
Kinetic energy + gravitational potential energy = a constant
When kinetic energy goes up by some amount, gravitational potential energy goes down by the same amount, and vice-versa

12. Orbital Energy and Speed
From energy calculations, we can determine the orbital speed of an object moving in a circular orbit about the Sun:
M = mass of Sun
d = distance of object from the Sun
For an elliptical orbit, the average orbital speed is about the same as the circular speed for a semimajor axis length that is the same as the radius (d) of the circular orbit
If the orbital speed of an object is large enough, it can escape from its orbit
This speed (escape velocity) is also the speed necessary to escape the influence of gravity of an object
Important for launching spacecraft from Earth
(circular speed) d a
(d = a)
(escape velocity)

13. Orbital Trajectories
Bodies moving slower than escape velocity follow circular or elliptical trajectory
Bodies moving at escape velocity follow a parabolic trajectory
Bodies moving faster than escape velocity follow a hyperbolic trajectory
Circle
Ellipse
Parabola
Hyperbola (

14. Tides
Whenever one object is attracted gravitationally to another, the various parts of the object feel gravitational forces differing in strength and direction
The differences in gravitational attraction that occur are called tidal forces
Strength of tidal force depends on strength of gravity on side nearest attractor vs. that on side farthest from attractor
Tides are distortions in shape resulting from tidal forces
Moon (m)
Earth (M)

15. Tides
Tidal distortions cause an elongation of Earth and the Moon along a line connecting the centers of both bodies
The effect of tides can be seen by noting the differences in acceleration (including direction) at various points on or in the object, measured relative to tidal acceleration at the center
Tidal acceleration has both horizontal and vertical components in general
Moon
Earth
(The amount of tidal distortion is exaggerated greatly in this drawing!)

16. Tides Both the Sun and Moon cause tidal forces on Earth
Because of the Sun’s much greater distance, the solar tidal acceleration is only about ½ of the lunar tidal acceleration
Since the Earth’s interior is very rigid, the solid Earth distorts very little
Maximum tidal distortion of solid Earth is only about 20 cm
Tidal forces cause water in oceans to flow
Produced by horizontal components of tidal forces
Tidal force is weak (one ten millionth as strong as gravity of the Earth)
Collective motions of water molecules produce a propagating wave that moves around Earth
High tide occurs just after Moon crosses the meridian and when Moon is on the opposite side of Earth (12 hours 25 minutes later)

17. Tides
Thus there are 2 high and 2 low tides per day at any given location
In a few areas on Earth (like the Gulf of Mexico), local effects of the water basin act to modify tidal forces, and these areas only have 1 high and 1 low tide per day
When the Sun and Moon are aligned (full or new Moon), the total tidal acceleration is the sum of the tidal accelerations from the Sun and Moon
Leads to what are called spring tides
High (low) tides are unusually high (low)
Has nothing to do with the spring season
At first or last quarter Moon, tidal accelerations of Moon and Sun partially cancel (neap tides)
Tidal acceleration at spring tide is about 3x larger than at neap tide

18. Tides Spring tide example Neap tide example Moon Sun Earth Moon Sun

19. General Relativity
Newton’s theory of gravity doesn’t work well when gravitational forces become too large (as is the case for celestial objects)
In this case, we need Albert Einstein’s General Theory of Relativity (1916), a new way to think about space and time
Newton’s theory of gravity is really just an approximation when distances, fields, and speeds are on the order of our everyday experiences

20. General Relativity
At the foundation of general relativity is the Principle of Equivalence:
There is no experiment that can be done in a small confined space that can detect the difference between a uniform gravitational field and an equivalent uniform acceleration.

21. General Relativity Consider the following situations:
Experiments performed in spacecraft accelerating in free space with a = – g will give same results
a = – g g
Experiments performed in spacecraft while at rest on Earth will show affects of constant acceleration of gravity g
 We can’t distinguish between the effects of an acceleration and gravity

22. General Relativity
The equivalence between accelerated motion and gravity suggested to Einstein a relationship between space, time, and gravity
According to general relativity, the presence of matter (and energy) causes “spacetime” to warp or curve
Motion of particles determined by shape of spacetime
Amount of curvature is proportional to the density of mass and energy  the more mass or energy, the larger the curvature
In the limit of speeds much smaller than the speed of light and weak gravitational forces, space is nearly flat and we can safely use Newton’s theory of gravity

23. General Relativity
M.A. Seeds, The Solar System, 5th Ed., Thomson/Brooks-Cole, 2007

24. General Relativity
An implication of curved spacetime is that light is affected by gravity
Light merely follows the contour of spacetime
Although effects of general relativity can be very small (at surface of Sun, spacetime curvature is only 1 part in 106), experimental tests have been performed which verify predictions of general relativity
Gravitational bending of starlight passing close to Sun measured during total eclipse of Sun in 1919
Two independent measurements in South America and Africa obtained 1.98  0.16 and 1.61  0.40 seconds of arc
General theory of relativity predicted 1.75 seconds of arc
Many more experimental verifications obtained since