Energy

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Energy

• What is it?
• Energy is conserved
• How do we use it in physics

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An Energy Pathway

• We need energy
• Where does it come from?
• Where does the energy in food come from?

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An Energy Pathway
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An Energy Pathway

• Where does the Energy of the Sun come from?

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Energy

• Energy can be transformed from one form to
  another. Nuclear-gt Heat and Light ?Chemical
  ?Mechanical
• What is conversion rate?
• We can account for every bit of energy along the
  way. ENERGY IS CONSERVED

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The Law of Conservation of Energy
The total energy of the Universe is unchanged by
any physical process.
Energy may be converted from one form to another
or transferred between bodies. It never
disappears. It cannot be created or destroyed.
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• Does the power generated
• affect the speed of the wind?
• Would locations behind the
• windmills have more wind if
• the windmills werent there?
• Yes
• No

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Energy

• Energy has a somewhat abstract nature. Try to be
  concrete.
• Energy is the property of a system that enables
  it to do work.
• What is work?

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Work by a Constant Force
Work is an energy transfer by the application of
a force. For work to be done there must be a
nonzero displacement. This is a bit
circular. Work Force x Distance if F and D are
parallel. If not use the component of F that
is parallel to displacement
The unit of work and energy is the joule (J). 1
J 1 Nm 1 kg m2/s2.
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Is he doing any work against the wall?
A) Yes B) No
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It is only the force in the direction of the
displacement that does work.
An FBD for the box at left
The work done by the force F is
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The work done by the normal force N is
The normal force is perpendicular to the
displacement.
The work done by gravity (w) is
The force of gravity is perpendicular to the
displacement.
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The net work done on the box is
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Fig. 06.08
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In general, the work done by a force F is defined
as
where F is the magnitude of the force, ?r is the
magnitude of the objects displacement, and ? is
the angle between F and ?r (drawn tail-to-tail).
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Example A ball is tossed straight up. What is
the work done by the force of gravity on the ball
as it rises?
FBD for rising ball
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Example A box of mass m is towed up a
frictionless incline at constant speed. The
applied force F is parallel to the incline. What
is the net work done on the box?
An FBD for the box
Apply Newtons 2nd Law
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Example What is the net work done on the box in
the previous example if the box is not pulled at
constant speed?
Proceeding as before
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Energy and Work

• We defined energy and work
• Review Work can be negative
• Machines help us do work
• Kinetic Energy
• Potential Energy

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Example A ball is tossed straight up. What is
the work done by the force of gravity on the ball
as it rises?
FBD for rising ball
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Question

• A box with mass m1 is being pulled up a rough
  incline by a rope-pulley-weight system.
• How many forces are doing work on the box?
• A) one force
• B) two forces
• C) three forces
• D) four forces
• E) no forces are doing
• work

Gravity, friction and tension
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Machines

• Machines help us do work.
• They can increase the Force at our disposal.
• We still have to do the same amount of work.
• Energy is conserved!!

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Fig. 06.03
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Fig. 06.10
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Kinetic Energy

• An object that is moving has the ability to do
  work

The faster the ball is moving the more the pins
will bounce. The heavier the ball the more the
pins will bounce
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Kinetic Energy
is an objects translational kinetic energy.
This is the energy an object has because of its
state of motion.
It can be shown that, in general
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Example The extinction of the dinosaurs and the
majority of species on Earth in the Cretaceous
Period (65 Myr ago) is thought to have been
caused by an asteroid striking the Earth near the
Yucatan Peninsula. The resulting ejecta caused
widespread global climate change.
If the mass of the asteroid was 1016 kg (diameter
in the range of 4-9 miles) and had a speed of
30.0 km/sec, what was the asteroids kinetic
energy?
This is equivalent to 109 Megatons of TNT.
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Gone with the Wind

• 20 mph winds hardly ever do damage
• 60 mph almost always do uproot trees, rip
  shingles off roofs. Why?
• 60 mph winds have 9 times the energy (1/2mv2) and
  can do 9 times as much (destructive ) work

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How much work do the brakes do when your car
decreases its speed from 25m/s to 20 m/s? Take m
1,500 kg
?K ½mvf2 - ½mvi2 ½m(vf2 vi2 )
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Question

• Compare the kinetic energy of two balls
• ball 1 mass m thrown with speed 2v
• ball 2 mass 2m thrown with speed v
• A) KE1 4KE2
• B) KE1 KE2
• C) 2KE1 KE2
• D) KE1 2KE2

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Question

• Your brakes can apply a constant force, F, to the
  wheels of your car. If it takes a distance
• of 30 m to stop when you are traveling 40 km/hr,
  what distance is required to stop when you are
  traveling 120 km/hr?
• A) 10 m
• B) 90 m
• C) 110 m
• D) 270 m

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Potential Energy

• Object can store energy because of its position.
  Examples
• A stretched or compressed spring
• A drawn bow
• Water in an elevated reservoir
• Chemical energy because of the position of
  electrons

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Gravitational Potential Energy
Objects have gravitational potential energy (PE)
because of their elevated position.
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Gravitational Potential Energy
Objects have gravitational potential energy (GPE)
because of their elevated position. Work is done
to lift an object and this is equal to the amount
of GPE. Lifting an object up to a height h at
constant v requires a Force of mg
F We can get this
back Work F?x mg ?x mgh GPE
mg This is minus the work done by gravity -Wg

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The change in gravitational potential energy
(only near the surface of the Earth) is
where ?y is the change in the objects vertical
position with respect to some reference point
that you are free to choose.
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GPE depends only on h not on how it got there
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Example What is the change in gravitational
potential energy of the box if it is placed on
the table? The table is 1.0 m tall and the mass
of the box is 1.0 kg.
First Choose the reference level at the floor. U
0 here.
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Example continued
Now take the reference level (U 0) to be on top
of the table so that yi ?1.0 m and yf 0.0 m.
The results do not depend on the location of U
0.
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Potential Energy
Objects have potential energy because of their
location (or configuration).
There are potential energies associated with
different (but not all!) forces. Such a force is
called a conservative force. Friction is not a
nonconservative .
In general
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Mechanical energy
Whenever nonconservative forces do no work, the
mechanical energy of a system is conserved. That
is Ei Ef or ?K ??U.
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Conservation of Mechanical Energy
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Conservation of Mechanical Energy
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Exam on Monday

• Covers chapters 4 5 (lectures posted on course
  website)
• Same format as first exam
• Equation sheet from Master the Concepts section
• Will post answers to Practice exam
• First exam I will add 5 pts to everyones grade.

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Energy

• Energy
• Work
• Kinetic Energy
• Potential Energy
• Conservation of Mechanical Energy
• KU constant
• More on Gravitational Potential Energy
• Variable forces
• Power

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Example (text problem 6.28) A cart starts from
position 4 with v 15.0 m/s to the left. Find
the speed of the cart at positions 1, 2, and 3.
Ignore friction.
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Example (text problem 6.84) A roller coaster car
is about to roll down a track. Ignore friction
and air resistance.
m 988 kg

• At what speed does the car reach the top of the
  loop?
• What is the force exerted on the car by the track
  at the top of the loop?

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Example continued
(b) What is the force exerted on the car by the
track at the top of the loop?
FBD for the car
Apply Newtons Second Law
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Example continued
(c) From what minimum height above the bottom of
the track can the car be released so that it does
not lose contact with the track at the top of the
loop?
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What do you do when there are nonconservative
forces? For example, if friction is present
The work done by friction.
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Dissipative Forces
Box is moving with constant v An FBD for the box
at left
constant v
f
f
Ftot ma 0 gt Fcos? f 0
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Dissipative forces

• The work done by friction is
• fx?x Fx?x
• We cannot retrieve this work as with gravity
• The work has gone into heating up the box and the
  ground
• Friction dissipated this amount of work

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Gravitational Potential Energy Part 2
We have considered GPE near the Earth The general
expression for gravitational potential energy is
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Example What is the gravitational potential
energy of a body of mass m on the surface of the
Earth?
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Example A planet of mass m has an elliptical
orbit around the Sun. The elliptical nature of
the orbit means that the distance between the
planet and Sun varies as the planet follows its
orbital path. How does the speed of a planet
vary as it orbits the Sun once. Take the planet
to move counterclockwise from B, its initial
location.
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Example A planet of mass m has an elliptical
orbit around the Sun. The elliptical nature of
the orbit means that the distance between the
planet and Sun varies as the planet follows its
orbital path. How does the speed of a planet
vary as it orbits the Sun once. Take the planet
to move counterclockwise from B its initial
location.
The mechanical energy of the planet-sun system
is
At the planet moves r decreases and the planet
moves faster. As the planet moves past point A
r begins to increase and the planet moves slower.

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Escape velocity

• What if energy of a rocket moving away from Earth
  is lt0?
• Can r be arbitrarily large?
• NO
• Rocket cannot get further than some rmax from
  Earth

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Escape velocity

• What if energy of the rocket is 0?
• Can r be arbitrarily large?
• YES if v? 0 as r ?8
• At Earths surface r R, the velocity needed to
  get arbitrarily far away is the escape velocity.

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Question

• Planet Xavier has a moon, Zeno, which orbits in a
  highly elliptical orbit? At which point does
  Zeno have maximum KE?

A) A B) B C) C D) Kinetic Energy is the same
at all points.
B
C
A
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Work by a Variable Force
Work is the area of the graph of Fx (the applied
force in the direction of the displacement)
versus the displacement. Remember x v?t area
under v vs. t graph
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Example What is the work done by the variable
force shown below?
The net work is then W1W2W3.
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Question

• Two forces are applied over a distance x as shown
  in the graph at right. F1 is constant
• over the distance, but F2 varies from zero to
  2F1. Compare the work done by the two
• forces.
• A) W1 gt W2
• B) W1 W2
• C) W1 lt W2

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Fig. 06.28
F
F -kx
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Hookes law.
For an ideal spring Fx ?kx
Fx is the magnitude of the force exerted by the
free end of the spring, x is the measured stretch
of the spring, and k is the spring constant (a
constant of proportionality its units are N/m).
A larger value of k implies a stiffer spring.
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Example (text problem 6.51) (a) If forces of 5.0
N applied to each end of a spring cause the
spring to stretch 3.5 cm from its relaxed length,
how far does a force of 7.0 N cause the same
spring to stretch?
For springs F?x. This allows us to write
Solving for x2
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Example continued
(b) What is the spring constant of this spring?
Or
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Example (text problem 6.48) An ideal spring has
k 20.0 N/m. What is the amount of work done
(by an external agent) to stretch the spring 0.40
m from its relaxed length?
Fx (N)
kx1
x (m)
x10.4 m
This is the elastic potential energy of a
spring It is analogous to mgh for GPE
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Elastic potential energy
The work done in stretching/compressing a spring
transfers energy to the spring.
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Example A box of mass 0.25 kg slides along a
horizontal, frictionless surface with a speed of
3.0 m/s. The box encounters a spring with k
200 N/m. How far is the spring compressed when
the box is brought to rest?
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Question

• A spring is compressed a distance x to give a
  rocket an initial velocity v. To double the
  initial speed of the rocket, the spring should be
  compressed a distance ______.
• A) 1/2 x
• B) x
• C) xÖ 2
• D) 2x
• E) 4x

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Power
Power is the rate of energy transfer.
Average Power
The unit of power is the watt. 1 watt 1 J/s
1 W.
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Power
Instantaneous Power
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Example (text problem 6.75) A race car with a
mass of 500.0 kg completes a quarter-mile (402 m)
race in a time of 4.2 s starting from rest. The
cars final speed is 125 m/s. What is the
engines average power output? Neglect friction
and air resistance.
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Question

• What power must an engine have if it is to be
  used to raise a 25-kg load 10 m in
• 4 seconds?
• A) 25 W
• B) 625 W
• C) 1000 W
• D) 2500 W

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Summary

• Conservation of Energy
• Calculation of Work Done by a Constant or
  Variable Force
• Kinetic Energy
• Potential Energy (gravitational, elastic)
• Power