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Key Concepts: Lecture 9
Newton
Newton’s Laws of Motion
Describing Motion
• Position
• Velocity
– Rate of change of position (speed & direction)
More on Kepler’s Laws
• Acceleration
– Rate of change of velocity
80 km/hr
40 km/hr
(a) Change in speed but not direction
Newton’s Law of Universal Gravitation
2 examples of acceleration:
(b) Change in direction but not speed
Newton 1642-1727
• Only child, posthumous son of an illiterate
yeoman
–
–
–
–
born prematurely - sickly as child
raised by maternal grandmother
as a child he built clocks & sundials
practical joker
• Trinity College, Cambridge University at 18
– studied mathematics & astrology
– encouraged to study physics by Barrow
• University closed in 1665 due to plague
– Invented calculus, studied gravity, optics
• Barrow resigns & gives Newton his post at
Cambridge
Newton’s Laws of Motion
• Law I: Law of Inertia
– A body at rest or in motion at a
constant velocity along a straight
line remains in that state of rest or
motion unless acted on by a net
outside force.
• Takes next logical step beyond
Galileo’s definition of inertia (tendency
of a body to keep moving after all forces stop
acting on it)
• Uniform motion is just as natural a
state for a body as being at rest
Laws of Motion
• Law II - The Force Law
– The acceleration (a) due to an applied
force (F) is in the same direction as the
force & is proportional to the strength
of the force & is inversely proportional
to the object’s mass (m)
• The units of force are chosen so the
constant is 1. So we write a = F/m
• To have acceleration there must be a force
• Force & acceleration always work in the
same direction
• Given the same force, a more massive
object accelerates more slowly than a less
massive one
Question?
• A ball is attached to a string and I spin it
abound my head in a circle
a∝F/m
a = constant x F /m
constant = 1
a = F/m
We can write this as
F=ma
– Is the ball accelerating?
– If it is accelerating what is the force?
– If the string were to break what path would the
ball follow?
Force=Mass xAcceleration
Examples of the Second Law
• Friction
– Hockey puck on ice vs. on a street
• Impact of a bat on a baseball
– The bat imparts a force to the ball and sends it
flying in the opposite direction
Laws of Motion
• Law III - The Reaction Law
• For every applied force, there is
an equal, but opposite force
• Forces always occur in pairs
• a force cannot be created in
isolation - need at least two bodies
acting against each other
• if gravity is a force it must act
between bodies
Newton Figures Out Gravity
Question?
• You push a cart and it moves but you do
not appear to move.
– Why don’t you move if there is an opposite
and equal force pushing on you?
• He unified the force which makes an apple
drop from a tree and the force which makes
the moon orbit the earth
– Gravity causes all objects to attract
one another
• He intuitively figured out that the force of
gravity between two objects depends on
only three things:
– The two masses of the objects: more
massive objects gives a stronger
attractive force
– The distance between the objects:
moving objects further apart weakens
the force
• This is true on size scales from a laboratory
desk to groups of stars and galaxies
Newton’s Law of Universal Gravitation
Demonstration: 2nd and 3rd of Newton’s Laws
• 2nd Law: Force = mass x acceleration (F = m a)
• 3rd Law: When two bodies interact the forces they
feel are opposite in direction and equal in strength.
• You push a cart and it moves but you do not appear
to move. Why? Because friction couples you to the
massive Earth, which recoils only a very tiny amount.
• If we remove friction, then we can see the two
motions more clearly…
Skate boards for Thur?
•Force is proportional to the
masses, m1 and m2. Smaller
masses→smaller force
•Force is inversely proportional to
the square of the distance
between the objects. Further
apart→ weaker force. This kind of
dependence of force with distance
is known as an “Inverse Square
Law.” Force weakens like the
square of the distance: if you
double the distance, the force
changes by a factor of 1/(2x2) =
1/4.
Orbits and Gravity
•
•
•
•
Gravity is the force which keeps the planets
from flying off into space
– Because the Sun is much more massive
then the planets the Sun controls the
motion of the planets
Gravity always pulls the planet toward the
Sun
Inertia wants to keep the planet moving in a
straight line
The balance between gravity and inertia
leads to the stable orbit of a planet
Throw the ball fast enough and it will go into orbit…
- The Moon is much closer to the Earth than the Sun.
- The inverse square law nature of gravity means that the Moon’s orbit is
controlled by the mass of the Earth, rather than the Sun.
- The force the Moon feels from the Earth is stronger than that it feels from
the Sun.
[Try and compare the ratio of the forces the Moon feels from the Sun and
the Earth to verify this.]
The Shapes of Orbits
•
The shape of an object’s orbit depends
on its velocity perpendicular to the force
of gravity
– A body with a small perpendicular
velocity will fall nearly straight in
– A body with a large perpendicular
velocity will overcome the force of
gravity and move to a larger
distance
•
For closed orbits the shapes will be
ellipses
•
Escape Velocity: if the velocity of an
object is greater than a certain value,
the escape velocity, then gravity is
unable to slow down the object enough
to prevent it from flying out to deep
space. For the Earth the escape velocity
is about 11 km/s.
Relation of Newton’s work to
Kepler’s 2nd Law
• As a planet moves toward the Sun the force
of gravity causes it to accelerate along its
orbits and it moves faster
• As a planet moves away from the Sun the
force of gravity acts along its orbit and
slows it down
Relation of Newton’s work to
Kepler’s 3rd Law
• Planets with larger average
distances from the Sun have
longer periods (P2=ka3)
– Since the gravitational
acceleration is less they move
more slowly along their orbits
– The orbits are larger
Note Kepler’s 3rd Law Applies to Any
Object Orbiting the Sun
(can be on near circular orbits or very elliptical, i.e. very
eccentric)
• P2 = k a3 , with k = constant, P = period of orbit,
and a = average distance of object from Sun
• Earth has P=1 year, a = 1 AU, so using these units,
k=1, and we can write P2=a3
• Examples
- If a planet has a = 4 AU, then it must have
period P = 8 years: 8x8 = 64 = 4x4x4
- This formula applies to any orbit around the Sun,
from circular to very eccentric (e.g. comets).
Newton’s Version of Kepler’s 3rd Law
• Newton applied his laws of motion and gravity to derive a
modified version of Kepler’s 3rd law (which was P2 = k a3 )
(Mass1+Mass2)x Period2 = K(average distance)3
• Here K is a new constant.
• This equation can also be written as Mtotal P2 = K a3
• In the solar system the mass of the Sun is so large that
Mtotal= Msun+Mplanet is almost exactly equal to Msun
• This law allows the determination of masses for distant
objects if the orbital properties (P and a) can be measured.
Newton’s Cosmology
• Gravity holds the solar system
together
– The Sun is the most massive object so its
gravity dominates the solar system
– The law of Universal Gravitation
naturally produces elliptical orbits
(Kepler’s 1st law)
– The law of Universal Gravitation
naturally produces Kepler’s 2nd and 3rd
laws
• Newton thought that beyond the solar
system, the universe of stars must be
infinite or it would collapse. We shall
see later if Newton was right.
Complexity to Simplicity
• For centuries people had tried to understand the
unique motions of the planets
1 - They were Gods that had special power over our lives
2 - They were mystical bodies moving in a complex clock
work universe with circular orbits, epicycles, and other
geometrical devices
• Not composed of the same material as the Earth
• Not covered by the same laws of nature as the Earth
3 - They were special objects moving under the control of
three laws of planetary motion (Kepler)
Complexity to Simplicity
4 - Laws
of physics (Motion and Gravitation) described the
motion of the planets & much, much more
– The planets obey the same laws of motion and gravity as any
object on the Earth or in the universe
– The planets are composed of the same types of matter as is the
Earth
– The same laws of motion & gravitation can explain a wide range
of phenomena
• The orbits of planets
• Tides
• How to build a bridge or tall building & land people on the Moon
[We will see later in the class that our understanding of the
physics laws has evolved further since Newton’s time due to
the work of Albert Einstein.]