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Energy

Energy is measured in Joules (J)

A sense for size:

Energy used to move a car 1 mile: 600 kJ

Energy to lift a plate to cupboard: about 1 J

Energy needed to lift me to the next floor:
about 1600 J
Work

Work: force taking
place over distance.

W = F * d

Work is only done
when an object
moves against a
force.

No move, no work.

No force, no work.
Example Problem

How much work do you do when you
push a crate with a force of 150 N two
meters?

Work = Force * distance so...
150 N * 2 m = 300 J
More Work

The force and the
displacement in work
must be in the same
direction.

If not we consider only the
component of
displacement aligned with
the force.

No matter how it reaches
2 meters from the floor,
the 1 N box has the same
work done on it due to
gravity.
Potential Energy

Energy in a stored state:

Objects suspended above the surface of
earth

stretched springs

chemical energy within gasoline
Gravitational Potential Energy

PE = m*g*h

m = mass

g = 10 m/s2

h = distance above
surface

Double m double PE

Double h double PE
Example Potential Energy Problem

Two cars, one twice as heavy as the
other, are lifted to the same elevation in a
service station. How do their potential
energies compare?

Potential energies are directly
proportional to mass; so a car of twice the
mass would have twice the potential
energy.
Kinetic Energy
• Energy of motion –
– KE = ½*m*v2
– So the more massive the more
energy
– but more dramatic, the more
velocity the MORE energy.
Potential Energy and Kinetic
Energy freely interconvert:
dropped objects, the release
of a bow, etc.
Example Kinetic Energy Problem
• Which has more kinetic energy – a car traveling at 30
km/hr or a car half the mass traveling at 60 km/hr?
• A little more complicated: KE depends directly on
mass so halving the mass would have the kinetic
energy BUT it depends on the square of the velocity
so twice the velocity would be 4 times the KE. The
result is that the lighter car at 60 km/hr has twice the
kinetic energy of the heavier, slower car.
Conversion of Energy from one form to
another
• At the top you have high
potential energy.
• As you fall, the amount
of potential energy falls,
but kinetic energy
increases.
• At the bottom all the
energy is converted to
kinetic energy.
So a pendulum is kinda interesting...
• All potential energy at either end.
• All kinetic at the lowest point.
• Knowing how far the bob falls you can calculate
exactly how quickly it will be going at the bottom.
Work-Energy Theorem
• When work is done on something it gains
kinetic energy.
• W = ΔKE
• If an object’s energy changes we know work
has been done on it.
Conservation of energy
• In the absence of external work input or output,
the energy of a system remains unchanged.
Energy cannot be created or destroyed.
Demo: Bowling Ball
• Energy is conserved so:
– Initial potential energy just enough to reach me
– Kinetic energy at bottom of path will approximately
equal that potential energy.
– Since energy can only be lost (air resistance) it
shouldn’t touch my face.
Gravity
Discovery of Gravity
People have long known that
most thing fall to earth. So
what was the ‘discovery’?
• The ancients believed that
everything had a natural
place and sought to return to
it.
• Newton, however, was the
first to think of falling objects
as being under the influence
of a force.
Orbits of heavenly bodies
The real breakthrough was in thinking of heavenly
bodies (the moon, planets, etc.) as being under the
influence of the same force as the apple.
Review: Projectile Motion
2
2
1
gtd =
• Imagine you throw the
object in a straight line
(no gravity).
• Due to gravity the object
will be found below this
line – by exactly the
same distance as if it
had been dropped.
The Moon “Falls” Around the Earth
• Newton imagined the moon
would travel in a straight line
– unless pulled downward by
Earth.
• Ultimately, he proved that,
just as the projectile falls
under its ideal path, so does
the moon.
• The force that makes an
apple fall and the moon orbit
are the same.
Gravitational Force
• The gravitational force is
always attractive.
• Proportional to each mass
involved (doubling m1 OR m2
doubles the force).
• Inversely proportional to
square of the distance
between masses (double d,
¼ force).
2
21
d
m*m*G
F =
Inverse Square Law
As distance from source increases, the area of a shell around the source
increases as the square of distance.
This results in the coverage (thickness) of the paint decreasing with the area.
9 16
1/9 1/16
Gravitational strength falls rapidly with
distance
• Sun weights 333,000
times more than earth.
• Sun is 24,000 times
further away (than the
center of the earth).
• Acceleration due to the
sun at earth’s surface:
0.006 m/s2
Example Problem
Imagine an impossible
ladder that extends into
the sky.
• A girl, weighing 600 N,
stands at the bottom. She
then climbs to four times
the earth’s radius.
– What is her weight now?
Distance goes from 1*earth radius to 4*earth radius (ladder adds 3x earth radius).
Distance therefore increases 4 times.
Distance squared will go up 16 times.
Force depends on the inverse of distance squared so it will go down 16 times:
600/16= 37.5 N
Possible Test Problem
• Consider a 1-N apple in a
tree.
– If the tree were twice as tall,
would the weight of the apple
be ¼ as much?
– Why or why not?
– The problem is one of “where
is our reference point?” Similar
to the problem with
temperature – We need to
count from the ‘right’ zero.
The right zero is the center of the earth, some 6000 Km (6,000,000 meters)
Increasing the height by 2 meters is negligible (6 million vs. 6 million and 2 meters).
Weight
• Weight only matches
gravitational force when
you are not accelerating.
• What you are actually
sensing is the normal
force supporting you.
N
m*g
m*a
m*g
• When you fall you accelerate.
• If your acceleration matches
gravitational acceleration (10
m/s2
) you will feel weightless.
• This does not mean gravity
is no longer acting on you.
– If gravity were no longer acting on
you then you would not
accelerate.
m*a
m*g
Weight and Gravity not necessarily
equal:
What do you feel inside an
elevator?
• Going up, as you accelerate
you feel heavy.
• Going down, as you
accelerate, you feel light.
• When you stop you feel
normal weight.
• (If the elevator were to break
and you were to drop you
would feel briefly
weightless).
Weightlessness
• If you accelerate freely
you have no weight.
• Freefall, no matter
where it occurs, results
in ‘no weight’.
• You may still be
undergoing a
gravitational
acceleration, such as an
astronaut in orbit.
Connection: Newton’s 2nd
Law
• Fnet = m*a
• When standing on earth Fnet = 0
N – m*g = 0
• When falling Fnet = m*g
m*g = m*g
• When accelerating upward Fnet > 0
N – m*g > 0 (so N > m*g)
• When accelerating downward Fnet < 0
N – m*g < 0 (so N < m*g)
Artificial Gravity
• When orbiting earth you feel weightless, this has
several consequences:
– The body begins to lose muscle mass as it does not need to
exert as much force to move.
– Some organs lose capacity.
• To combat this, several methods of producing gravity
have been proposed:
– For long-distance travel, have the spacecraft accelerate
slightly the whole trip.
– For orbital stations, take advantage of rotation.
• centripetal acceleration: a = v2
/r
Summing Up
• Gravitational force:
– double m1 or m2, double force
• What if you double both?
– double d, ¼ force.
• Weightlessness:
– Weight depends upon having a support force.
• If everything falls together, as on the space shuttle, no support force
needed.
– When undergoing an acceleration the support force may be
larger or smaller than gravitational force.
2
21
d
m*m*G
F =

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Lecture 9

  • 2.  Energy is measured in Joules (J)  A sense for size:  Energy used to move a car 1 mile: 600 kJ  Energy to lift a plate to cupboard: about 1 J  Energy needed to lift me to the next floor: about 1600 J
  • 3. Work  Work: force taking place over distance.  W = F * d  Work is only done when an object moves against a force.  No move, no work.  No force, no work.
  • 4. Example Problem  How much work do you do when you push a crate with a force of 150 N two meters?  Work = Force * distance so... 150 N * 2 m = 300 J
  • 5. More Work  The force and the displacement in work must be in the same direction.  If not we consider only the component of displacement aligned with the force.  No matter how it reaches 2 meters from the floor, the 1 N box has the same work done on it due to gravity.
  • 6. Potential Energy  Energy in a stored state:  Objects suspended above the surface of earth  stretched springs  chemical energy within gasoline
  • 7. Gravitational Potential Energy  PE = m*g*h  m = mass  g = 10 m/s2  h = distance above surface  Double m double PE  Double h double PE
  • 8. Example Potential Energy Problem  Two cars, one twice as heavy as the other, are lifted to the same elevation in a service station. How do their potential energies compare?  Potential energies are directly proportional to mass; so a car of twice the mass would have twice the potential energy.
  • 9. Kinetic Energy • Energy of motion – – KE = ½*m*v2 – So the more massive the more energy – but more dramatic, the more velocity the MORE energy. Potential Energy and Kinetic Energy freely interconvert: dropped objects, the release of a bow, etc.
  • 10. Example Kinetic Energy Problem • Which has more kinetic energy – a car traveling at 30 km/hr or a car half the mass traveling at 60 km/hr? • A little more complicated: KE depends directly on mass so halving the mass would have the kinetic energy BUT it depends on the square of the velocity so twice the velocity would be 4 times the KE. The result is that the lighter car at 60 km/hr has twice the kinetic energy of the heavier, slower car.
  • 11. Conversion of Energy from one form to another • At the top you have high potential energy. • As you fall, the amount of potential energy falls, but kinetic energy increases. • At the bottom all the energy is converted to kinetic energy.
  • 12. So a pendulum is kinda interesting... • All potential energy at either end. • All kinetic at the lowest point. • Knowing how far the bob falls you can calculate exactly how quickly it will be going at the bottom.
  • 13. Work-Energy Theorem • When work is done on something it gains kinetic energy. • W = ΔKE • If an object’s energy changes we know work has been done on it.
  • 14. Conservation of energy • In the absence of external work input or output, the energy of a system remains unchanged. Energy cannot be created or destroyed.
  • 15. Demo: Bowling Ball • Energy is conserved so: – Initial potential energy just enough to reach me – Kinetic energy at bottom of path will approximately equal that potential energy. – Since energy can only be lost (air resistance) it shouldn’t touch my face.
  • 17. Discovery of Gravity People have long known that most thing fall to earth. So what was the ‘discovery’? • The ancients believed that everything had a natural place and sought to return to it. • Newton, however, was the first to think of falling objects as being under the influence of a force.
  • 18. Orbits of heavenly bodies The real breakthrough was in thinking of heavenly bodies (the moon, planets, etc.) as being under the influence of the same force as the apple.
  • 19. Review: Projectile Motion 2 2 1 gtd = • Imagine you throw the object in a straight line (no gravity). • Due to gravity the object will be found below this line – by exactly the same distance as if it had been dropped.
  • 20. The Moon “Falls” Around the Earth • Newton imagined the moon would travel in a straight line – unless pulled downward by Earth. • Ultimately, he proved that, just as the projectile falls under its ideal path, so does the moon. • The force that makes an apple fall and the moon orbit are the same.
  • 21. Gravitational Force • The gravitational force is always attractive. • Proportional to each mass involved (doubling m1 OR m2 doubles the force). • Inversely proportional to square of the distance between masses (double d, ¼ force). 2 21 d m*m*G F =
  • 22. Inverse Square Law As distance from source increases, the area of a shell around the source increases as the square of distance. This results in the coverage (thickness) of the paint decreasing with the area. 9 16 1/9 1/16
  • 23. Gravitational strength falls rapidly with distance • Sun weights 333,000 times more than earth. • Sun is 24,000 times further away (than the center of the earth). • Acceleration due to the sun at earth’s surface: 0.006 m/s2
  • 24. Example Problem Imagine an impossible ladder that extends into the sky. • A girl, weighing 600 N, stands at the bottom. She then climbs to four times the earth’s radius. – What is her weight now? Distance goes from 1*earth radius to 4*earth radius (ladder adds 3x earth radius). Distance therefore increases 4 times. Distance squared will go up 16 times. Force depends on the inverse of distance squared so it will go down 16 times: 600/16= 37.5 N
  • 25. Possible Test Problem • Consider a 1-N apple in a tree. – If the tree were twice as tall, would the weight of the apple be ¼ as much? – Why or why not? – The problem is one of “where is our reference point?” Similar to the problem with temperature – We need to count from the ‘right’ zero. The right zero is the center of the earth, some 6000 Km (6,000,000 meters) Increasing the height by 2 meters is negligible (6 million vs. 6 million and 2 meters).
  • 26. Weight • Weight only matches gravitational force when you are not accelerating. • What you are actually sensing is the normal force supporting you. N m*g m*a m*g
  • 27. • When you fall you accelerate. • If your acceleration matches gravitational acceleration (10 m/s2 ) you will feel weightless. • This does not mean gravity is no longer acting on you. – If gravity were no longer acting on you then you would not accelerate. m*a m*g
  • 28. Weight and Gravity not necessarily equal: What do you feel inside an elevator? • Going up, as you accelerate you feel heavy. • Going down, as you accelerate, you feel light. • When you stop you feel normal weight. • (If the elevator were to break and you were to drop you would feel briefly weightless).
  • 29. Weightlessness • If you accelerate freely you have no weight. • Freefall, no matter where it occurs, results in ‘no weight’. • You may still be undergoing a gravitational acceleration, such as an astronaut in orbit.
  • 30. Connection: Newton’s 2nd Law • Fnet = m*a • When standing on earth Fnet = 0 N – m*g = 0 • When falling Fnet = m*g m*g = m*g • When accelerating upward Fnet > 0 N – m*g > 0 (so N > m*g) • When accelerating downward Fnet < 0 N – m*g < 0 (so N < m*g)
  • 31. Artificial Gravity • When orbiting earth you feel weightless, this has several consequences: – The body begins to lose muscle mass as it does not need to exert as much force to move. – Some organs lose capacity. • To combat this, several methods of producing gravity have been proposed: – For long-distance travel, have the spacecraft accelerate slightly the whole trip. – For orbital stations, take advantage of rotation. • centripetal acceleration: a = v2 /r
  • 32. Summing Up • Gravitational force: – double m1 or m2, double force • What if you double both? – double d, ¼ force. • Weightlessness: – Weight depends upon having a support force. • If everything falls together, as on the space shuttle, no support force needed. – When undergoing an acceleration the support force may be larger or smaller than gravitational force. 2 21 d m*m*G F =