Conservation of Energy

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Chapter 6

• Conservation of Energy

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Conservation of Energy

• Work by a Constant Force
• Kinetic Energy
• Potential Energy
• Work by a Variable Force
• Springs and Hookes Law
• Conservation of Energy
• Power

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The Law of Conservation of Energy
The total energy of the Universe is unchanged by
any physical process.
The three kinds of energy are kinetic energy,
potential energy, and rest energy. Energy may
be converted from one form to another or
transferred between bodies.
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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.
The unit of work and energy is the joule (J). 1
J 1 Nm 1 kg m2/s2.
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Work - Example
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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Work - Example
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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Work - Example
The net work done on the box is
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Work Done
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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Work - Example
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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Inclined Plane-V Constant
A box of mass m is towed up a frictionless
incline at constant speed. The applied force F
is parallel to the incline. Question What is
the net work done on the box?
An FBD for the box
Apply Newtons 2nd Law
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Inclined Plane-V Constant
Example continued
The magnitude of F is
If the box travels along the ramp a distance of
?x the work by the force F is
The work by gravity is
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Inclined Plane-V Constant
Example continued
The work by the normal force is
The net work done on the box is
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Inclined Plane-Acceleration
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
New Term
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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 Net Work
Change in K
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Kinetic Energy
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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Gravitational Potential Energy Part 1- Close to
Earths Surface
Potential energy is an energy of position.
There are potential energies associated with
different forces. Forces that have a potential
energy associated with them are called
conservative forces.
In general
Not all forces are conservative, i.e. Friction.
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Gravitational Potential Energy
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. You are free to choose to location of this
where ever it is convenient.
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GPE Problem
The table is 1.0 m tall and the mass of the box
is 1.0 kg. Ques What is the change in
gravitational potential energy of the box if it
is placed on the table?
U 0
First Choose the reference level at the floor. U
0 here.
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GPE Problem
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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Total Mechanical Energy
Mechanical energy is
The total mechanical energy of a system is
conserved whenever nonconservative forces do no
work. That is Ei Ef or ?K ??U.
Then if ?K increases ?U decreases and vice versa
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Mechanical Energy Problem
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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Mechanical Energy Problem
Or use E3E2
Or use E3E1 E2E1
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Roller Coaster Problem
A roller coaster car is about to roll down a
track. Ignore friction and air resistance.
m 988 kg
(a) At what speed does the car reach the top of
the loop?
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Roller Coaster Problem
Example continued
(b) What is the force exerted on the car by the
track at the top of the loop?
Apply Newtons Second Law
FBD for the car
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Roller Coaster Problem
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?
Using conservation of mechanical energy
Solve for the starting height
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Roller Coaster Problem
Example continued
v vmin when N 0. This means that
What is vmin?
The initial height must be
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Nonconservative Forces
What do you do when there are nonconservative
forces? For example, if friction is present
The work done by friction.
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Gravitational Potential Energy Part 2 - Away
from Earths Surface
The general expression for gravitational
potential energy is
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Gravitational Potential Energy
Example What is the gravitational potential
energy of a body of mass m on the surface of the
Earth?
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Planetary Motion
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. Take the planet to move counterclockwise
from its initial location. QUES How does the
speed of a planet vary as it orbits the Sun once?

The mechanical energy of the planet-sun system
is
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Planetary Motion
At point B the planet is the farthest from the
Sun. At point A the planet is at its closest
approach to the sun.
Starting from point B (where the planet moves
the slowest), as the planet moves in its orbit r
begins to decrease. As it decreases the planet
moves faster. At point A the planet reaches
its fastest speed. As the planet moves past point
A in its orbit, r begins to increase and the
planet moves slower.
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Work by a Variable Force
Work can be calculated by finding the area
underneath a plot of the applied force in the
direction of the displacement versus the
displacement.
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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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Spring Force
By hanging masses on a spring we find that
stretch ? applied force. This is 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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Spring Force
(a) A force of 5.0 N applied to the end of a
spring cause the spring to stretch 3.5 cm from
its relaxed length. Ques 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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Spring Force
Example continued
(b) What is the spring constant of this spring?
Or
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Spring Force
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?
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Elastic Potential Energy
The work done in stretching or compressing a
spring transfers energy to the spring. Below is
the equation of the spring potential energy.
The spring is considered the system
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Elastic Potential Energy
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.
Ques How far is the spring compressed when the
box is brought to rest?
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Power
Power is the rate of energy transfer.
Average Power
Instantaneous Power
The unit of power is the watt. 1 watt 1 J/s
1 W.
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Power - Car Example
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. (Neglect friction and air resistance.)
Ques What is the engines average power output?

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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