Download Newton`s laws, Motion and Gravity Newton`s laws, Motion and Gravity

Newton’s laws, Motion and Gravity
Reading: Chapter 3.5
Newton’s laws, Motion and Gravity
Questions
• How do we describe motion ?
• What are Newton’s laws ?
• What is the Theory of Gravitation ?
• How did Newton explain Kepler’s laws ?
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Describing Motion: Position and Time
Position and time specify where and when a particle is.
Position is measured relative to a coordinate system.
A coordinate system is an artificial grid that has an origin
(starting point) and an orientation.
Describing Motion: speed and velocity
• Speed: Rate at which an object moves (changes position)
speed = distance = !x
time
!t
Standard International (SI) units are m/s
example: speed of 10 m/s
Use SI units unless otherwise asked
• Velocity: Speed and direction
example: 10 m/s, due east
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Example
Car A moves 60 km east in 2 hours. Car B moves 60 km
south in 2 hours. Find their speeds and velocities in km/hr
1. Distance:
Car A: !x = 60 km
Car B: !x = 60 km
N
2. Direction:
Car A: east
Car B: south
!
vA
!
vB
3. Time Interval:
Car A: !t = 2hrs
Car B: !t = 2hrs
Example
Car A moves 60 km east in 1 hour. Car B moves 60 km
south in 1 hour.
4. Speed:
Car A = Car B:
!x 60km
=
= 30km / hr
!t 2hrs
5. Velocity:
Car A: 30 km/hr east
Car B: 30 km/hr south
N
!
vA
!
vB
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Describing Motion: Acceleration
Velocity can change if:
1. The speed changes
2. The direction changes
3. Both speed and direction changes
Acceleration is the rate of change of velocity with time.
acceleration = change of velocity/time
!
! !v
a=
!t
SI units:
m/s m
= 2
s
s
Since velocity has a direction,
acceleration also has a direction
The Acceleration of Gravity
• All falling objects have
constant acceleration
downwards (not
counting friction of air
resistance).
• On Earth, acceleration
of gravity, g ! 10 m/s2:
speed increases 10 m/s
with each second of
falling.
g = 9.81m / s 2
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The Acceleration of Gravity (g)
• Galileo showed that g is
the same for all falling
objects, regardless of
their mass.
• The motion of an object
solely under the influence
of gravity is called free
fall.
(Objects dropped or
thrown up vertically,
projectile motion,
orbiting satellites)
Apollo 15 demonstration
Check out
http://nssdc.gsfc.nasa.gov/
planetary/lunar/apollo_15_feather_drop.html
The Acceleration of Gravity
Example: Maya tosses a ball vertically up in the air with an
initial speed of 30 m/s. How long will it take for the ball to reach
its highest point ?
!
v
!
a
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The Acceleration of Gravity
Example: Maya tosses a ball vertically up in the air with an
initial speed of 30 m/s. How long will it take for the ball to reach
its highest point ?
Acceleration of gravity acts downwards, causing the ball to slow
down and finally stop before it falls downwards.
Initial velocity: 30 m/s (upwards)
!
v
Final velocity (at highest point): 0 m/s
Acceleration
!9.81m / s 2
(downwards)
!
a
Every second the ball will slow down by roughly 10 m/s. It will
therefore reach its highest point after 3 seconds
Describing Motion: Momentum
Momentum = mass x velocity
Units of momentum: units of mass x unit of velocity
SI (Standard International) unit of mass: kg
SI units of velocity: m/s
Hence SI unit of momentum: kg m/s
Since velocity has direction, momentum has direction.
Newton discovered a relationship between force and
momentum….
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Forces
A force is a push or a pull that acts on an object.
A force is exerted on an object. (An object cannot push
or pull on itself).
A force on an object is exerted by an agent. (The object
can apply a force back on the agent).
A force can be either
a) A contact force (the agent touches the object)
b) A long range force (agent does not physically touch
the object). (Example : gravity, magnetism).
Newton’s Laws of Motion
What are Newton’s three laws of motion?
Newton’s first law of motion:
An object remains at rest or at
constant velocity unless a net
force acts to change its speed
or direction.
Check out:
http://www.youtube.com/watch?v=7gfWvexu4u4&feature=related
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Newton’s Laws of Motion
What are Newton’s three laws of motion?
Newton’s first law of motion:
An object remains at rest or at
constant velocity unless a net
force acts to change its speed
or direction.
Note: If an object is at rest or moving at constant velocity this does
NOT mean that there are no forces acting on it. It means that the
SUM of all the forces in all directions is 0.
Thought Question:
Is there a net force? Y/N
1.
2.
3.
4.
5.
A car coming to a stop.
A bus speeding up.
An elevator moving up at constant speed.
A bicycle going around a curve.
A moon orbiting Jupiter.
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Thought Question:
Is there a net force? Y/N
1.
2.
3.
4.
5.
A car coming to a stop. Y
A bus speeding up. Y
An elevator moving at constant speed. N
A bicycle going around a curve. Y
A moon orbiting Jupiter. Y
Newton’s Laws of Motion
Newton’s second law of motion
Force = rate of change of momentum with time
!
! !(mv)
F=
!t
If the mass of an object remains constant,
!
!
!v
!
F=m
= ma
!t
Force = mass " acceleration
Units: Newtons
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Newton’s Laws of Motion
Newton’s third law of motion:
For every force, there is always an equal and
opposite reaction force.
Can you explain the motion
of the rocket using Newton’s
third law?
Thought Question:
Is the force the Earth exerts on you larger, smaller,
or the same as the force you exert on it?
A. Earth exerts a larger force on you.
B. I exert a larger force on Earth.
C. Earth and I exert equal and opposite forces on
each other.
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Thought Question:
Is the force the Earth exerts on you larger, smaller,
or the same as the force you exert on it?
A. Earth exerts a larger force on you.
B. I exert a larger force on Earth.
C. Earth and I exert equal and opposite forces
on each other.
Newton’s Theory of Gravity
The Universal Law of Gravitation:
1. Every mass attracts every other mass.
2. Attraction is directly proportional to the product of their
masses. (The more massive, the more attractive!)
3. Attraction is inversely proportional to the square of the
distance between their centers.
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Newton’s Theory of Gravity
F12 = F21 =
Gm1m2
d2
The gravitational force between two objects is typically very
small unless the objects are massive (example planets).
Example: Force between two people standing a metre apart:
F12 = F21 =
Gm1m2 (6.67 !10"11 Nm 2 / kg 2 )(60kg)(60kg)
=
= 2.4 !10"7 N
r2
(1m) 2
Example: Force between the earth and 1kg on the surface
F12 = F21 =
Gm1m2 (6.67 !10"11 Nm 2 / kg 2 )(M earth )(1kg)
=
= 9.8N = mg!
r2
(Rearth ) 2
Newton’s Theory of Gravity
Why do all objects fall at the same rate regardless of
their mass ?
•
According to the law of gravitation, force of gravity between the
Earth and an object of mass m:
F=
•
Gm mEarth
d2
According to Newtons 2nd law, this force can also be written as
F = ma
Equating the two, we see that the acceleration does not depend on the
objects’ mass
Gm mEarth
GmEarth
= ma ! a =
2
d
d2
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Newton’s Theory of Gravity
Finding the mass of the earth
By measuring G, the radius of the earth and g at the earth’s
surface we can obtain the mass of the earth without
actually ‘weighing’ it:
g=
mearth =
Gmearth
d2
2
gRearth
= 5.98 ! 10 24 kg
G
Mass versus weight
In physics mass and weight are related but are NOT the same
thing.
Mass is the amount of matter in an object.
Mass is an intrinsic property of an object.
Mass is the same no matter what forces are acting on an object.
When you report your weight at the doctors office as 60kg you
are actually reporting your mass….
International Prototype Kilogram (IPK)
Measuring mass
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Mass versus weight
In physics mass and weight are related but are NOT the same
thing.
Weight is the force of gravity exerted on an object.
Weight is directly proportional to the mass.
!
!
W = ma,
!
a : acceleration due to gravity
Example: Doubling your mass doubles your weight.
Weight varies from planet to planet since the acceleration due to
gravity varies.
Measuring weight
Thought Question
On the Moon:
A.
B.
C.
D.
My weight is the same, my mass is less.
My weight is less, my mass is the same.
My weight is more, my mass is the same.
My weight is more, my mass is less.
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Thought Question
On the Moon:
A.
B.
C.
D.
My weight is the same, my mass is less.
My weight is less, my mass is the same.
My weight is more, my mass is the same.
My weight is more, my mass is less.
Tides: Gravity in Action
• Tides are caused by small differences in
gravitational forces.
– As the Earth and moon orbit around each other, they
attract each other gravitationally.
– Because the side of Earth toward the moon is a bit
closer, the moon pulls on it more strongly and that
pulls up a bulge.
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Tides: Gravity in Action
• Also, the moon pulls on Earth a bit more than it
pulls on Earth’s far side and that produces a
bulge on the far side.
• Because there are two bulges, there are two
high tides each day
Orbital Motion
• An object orbiting Earth is actually falling
(being accelerated) toward Earth’s center.
– An object in a stable orbit
continuously misses Earth
because of its orbital
velocity.
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Apparent weight
Why are astronauts weightless in space?
• There is gravity in
space
• Weightlessness is
due to a constant
state of free-fall
around the earth
• The apparent
weight of an object
in freefall is zero (no
sensation of weight).
Apparent weight
Why are astronauts weightless in space?
Check out Dr. Stephen Hawking’s ‘weightless’ flight:
http://www.youtube.com/watch?v=OhIpdSZQZlI
Hawking in Waterloo, 2010
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Orbital Motion
• When the captain of a spaceship says to the
pilot, “Put us into a standard orbit,” the ship’s
computers calculate the velocity needed to
achieve a circular orbit.
– That circular velocity depends only on the mass of the
planet and the distance from the center of the planet.
Orbital Motion
• Once the ship reaches circular velocity, the
engines can shut down.
– The ship is in orbit and will fall around the planet
forever—so long as it is above the atmosphere’s friction.
– No further effort is needed to maintain orbit—thanks to the
laws Newton discovered.
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Escape Velocity
• If an object gains enough
orbital energy, it may escape
(change from a bound to
unbound orbit)
• Escape velocity from Earth !
11 km/s from sea level (about
40,000 km/hr)
Escape and orbital velocities don’t depend on the mass
of the object
Kepler’s Laws Revisited
Kepler’s First Law: The orbit of each planet around
the Sun is an ellipse with the Sun at one focus.
• Newton showed that the
inverse square law results in
elliptical orbits for the
planets.
• According to Newton’s 1st
law of motion the planets will
remain in their orbits unless
acted on by another force
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Kepler’s Laws Revisited
Kepler’s Second Law: As a planet moves around
its orbit, it sweeps out equal areas in equal times.
• The inverse square law
means that when planets
are closer to the sun they
feel a larger force and
thus move faster
Kepler’s Laws Revisited
Kepler’s Second Law: As a planet moves around
its orbit, it sweeps out equal areas in equal times.
• The total angular (orbital)
momentum, which is
proportional to the distance
from the sun and the velocity
conserved.
• Thus if the distance
increases, velocity decreases
and vice versa, to keep
angular momentum constant.
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Kepler’s Laws Revisited
Kepler’s Second Law: As a planet moves around
its orbit, it sweeps out equal areas in equal times.
• You can observe
conservation of angular
momentum for yourself.
• When a skater increases the
radius of her obit by raising
her arms, she decreases her
velocity and vice versa.
Kepler’s Laws Revisited
Kepler’s Third Law More distant planets orbit the Sun at
slower average speeds, obeying the relationship
p2 = a3
p = orbital period in years
a = avg. distance from Sun in AU
Newton derived a more general version:
p2 =
4! 2
a3
G(M1 + M 2 )
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Kepler’s Laws Revisited
p2 =
4! 2
a3
G(M1 + M 2 )
Newton’s laws of gravity and motion showed that
the relationship between the orbital period and
average orbital distance of a system tells us the total
mass of the system. This extended Kepler’s 3rd law
Examples:
• Earth’s orbital period (1 year) and average distance (1 AU)
tell us the Sun’s mass.
• Orbital period and distance of a satellite from Earth tell us
Earth’s mass.
• Orbital period and distance of a moon of Jupiter tell us
Jupiter’s mass.
Newton’s Universe
• The laws of motion and gravity were general laws that
described the motions of all bodies in the universe under the
action of external forces.
• In addition, the laws were predictive because they made
possible specific calculations of predictions that could be
tested by observation.
• Newton’s discoveries remade astronomy into an analytical
science.
– Astronomers could measure the positions and motions of
celestial bodies, calculate the gravitational forces acting on
them, and predict their future motion.
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Summary
• How do we describe motion ?
– Speed = distance / time
– Speed & direction => velocity
– Change in velocity/time => acceleration
– Momentum = mass x velocity
– Force causes change in momentum, producing
acceleration
Summary
• What are Newton’s Laws of Motion ?
– 1. Object remains at rest or at constant velocity
if no net force is acting.
– 2. Force = rate of change of momentum
If the mass is constant,
Force = mass " acceleration
– 3. For every force there is an equal and opposite
reaction force
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Summary
• How is mass different from weight ?
– Mass = quantity of matter
– Weight = force acting on mass
– Objects are weightless in free-fall
Summary
How did Newton explain Kepler’s laws ?
Newton formulated the Theory of Gravitation:
F12 = F21 =
Gm1m2
d2
The theory of gravity explains and extends Keplers laws:
- All objects obey Kepler’s laws, not just planets
- Orbits can be ellipses, parabolas or hyperbolas
- Objects orbit around their centre of mass
- Orbits can be used to predict mass of an object
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