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

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)

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)

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

mvp

maF net

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

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)

a

a

a

a

v

v

v

v

F

F

F

F

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 Nm/kg2

• M, m are the 2 interacting masses• d = distance between M and m

– Direction always indicates attraction• Equal magnitude forces in opposite directions

2d

GMmFG

M

m

FG

FG

d

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/s2

(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)

Mm

d

FG FG =

(spherically symmetric masses)

d

FG FG

(point masses concentrated at centers)

M m

mgR

GmMW

2Earth

(R = distance from Earth’s center)

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

2

1

r

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

MmG

aP

32 4

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

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

d

GMvc (circular speed)

d

GMve

2

(escape velocity)

(d = a)

d

a

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

(http://www.math2.org/math/algebra/conics.htm)

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

2d

GMmFG

Earth (M)Moon (m)

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

Earth

Moon

(The amount of tidal distortion is exaggerated greatly in this drawing!)

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)

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

Tides• Spring tide example

• Neap tide example

EarthSun

Moon

EarthSun

Moon

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

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.

General Relativity• Consider the following situations:

Experiments performed in spacecraft while at rest on Earth will show affects of constant acceleration of gravity g

g

Experiments performed in spacecraft accelerating in free space with a = – g will give same results

a = – g

We can’t distinguish between the effects of an acceleration and gravity

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

General Relativity

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

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