Section 5-3: Conservation of energy

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Section 5-3 Conservation of energy

• Objectives
• Identify situations in which conservation of
  mechanical energy is valid.
• Recognize the forms that conserved energy can
  take.
• Solve problems using conservation of mechanical
  energy.

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Mechanical Energy

• Mechanical energy is the energy that is possessed
  by an object due to its motion or due to its
  position. Mechanical energy can be either kinetic
  energy (energy of motion) or potential energy
  (stored energy of position) or both.

The total amount of mechanical energy is merely
the sum of the potential energy and the kinetic
energy
TME KE PEg PEs
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Mechanical Energy as the Ability to Do Work

• Any object that possesses mechanical energy -
  whether it is in the form of potential energy or
  kinetic energy - is able to do work.

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Conservative vs. non - conservative Forces

• There are a variety of ways to categorize all the
  types of forces.
• Contact force Forces that arise from the
  physical contact of two objects.
• Field force exist between objects, even in the
  absence of physical contact between the objects.
• We can also categorize forces based upon whether
  or not their presence is capable of changing an
  object's total mechanical energy.
• Conservative force can never change the total
  mechanical energy of an object
• Non-conservative forces will change the total
  mechanical energy of the object

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• The conservative forces include the gravity
  forces, spring force, magnetic force, electrical
  force.
• We will simply say all the other forces are
  non-conservative forces, such as applied force,
  normal force, tension force, friction force, and
  air resistance force.

conservative forces Non-conservative forces
Fgrav Fspring Fapp Ffrict Ften FNorm
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Work energy theorem
Wnet ?KE
Wgrav Wspring Wother ?KE
Wgrav work done by Gravity Wgrav mg(hi hf)
Wspring work done by spring Wspring ½ k(xi2
xf2)
mghi mghf ½ kxi2 ½kxf2 Wother ½ kvf2
½kvi2
mghi ½ kxi2 ½kvi2 Wother ½ kvf2 mghf
½kxf2
PEgi PEsi KEi Wother PEgf PEsf KEf
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PEgi PEsi KEi Wother PEgf PEsf KEf
TMEi Wother TMEf
Wother TMEf - TMEi

• When net work is done upon an object by an
  non-conservative force, the total mechanical
  energy (KE PE) of that object is changed.
• If the work is positive, then the object will
  gain energy.
• If the work is negative, such as friction doing
  work, then the object will lose energy, the
  object will gain heat (internal energy).
• The gain or loss in energy can be in the form of
  potential energy, kinetic energy, or both.
• The work done will be equal to the change in
  mechanical energy of the object.

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When Wother 0 TMEi TMEf

• When the only type of force doing net work upon
  an object is conservative force (Wother 0), the
  total mechanical energy (KE PE) of that object
  remains constant. TMEf TMEi. In such cases,
  the object's energy changes form.
• For example, as an object is "forced" from a high
  elevation to a lower elevation by gravity, some
  of the potential energy of that object is
  transformed into kinetic energy. Yet, the sum of
  the kinetic and potential energies remain
  constant.

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The Example of Pendulum Motion

• Consider a pendulum bob swinging to and fro on
  the end of a string. There are only two forces
  acting upon the pendulum bob. Gravity (an
  internal force) acts downward and the tensional
  force (an external force) pulls upwards towards
  the pivot point. The external force does not do
  work since at all times it is directed at a
  90-degree angle to the motion. The only force
  doing work is gravity, which is a conservative
  force.

KEi PEi Wext KEf PEf
Wext 0
KEi PEi KEf PEf
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The pendulum Wother 0

• The sum of the kinetic and potential energies in
  system is called the total mechanical energy.
• In the case of a pendulum, the total mechanical
  energy (KE PE) is constant at the highest
  point, all the energy is potential energy, at the
  lowest point, all the energy is kinetic energy.

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• As the 2.0-kg pendulum bob in the above diagram
  swings to and fro, its height and speed change.
  Use energy equations and the above data to
  determine the blanks in the above diagram.

0.153
0
0.306
0.306
1.73
2.45
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Roller coaster friction is ignored, Wother 0

• A roller coaster operates on the principle of
  energy transformation. Work is initially done on
  a roller coaster car to lift the car to the first
  and highest hill. The roller coaster car has a
  large quantity of potential energy and virtually
  no kinetic energy as it begins the trip down the
  first hill. As the car descents hills and loops,
  it potential energy is transformed into kinetic
  energy as the car ascends hills and loops, its
  kinetic energy is transformed into potential
  energy. The total mechanical energy of the car is
  conserved when friction is ignored.

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The skier

• Transformation of energy from the potential to
  the kinetic also occurs for a ski jumper. As a
  ski jumper glides down the hill towards the jump
  ramp and off the jump ramp towards the ground,
  potential energy is transformed into kinetic
  energy. If friction can be ignored, the total
  mechanical energy is conserved.

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A free falling object

• If a stationary object having mass m is located a
  vertical distance h above Earths surface, the
  object has initial PE mgh and KE 0. As object
  falls, its PE decreases and KE increases. The
  total mechanical energy is conserved.

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Energy conversion of a free falling object
The graph shows as a ball is dropped, how its
energy is transformed.

• The total mechanical energy remains
  _____________.
• GPE decreases as KE increases

constant
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Example 1

• A 55.0-kilogram diver falls freely from a diving
  platform that is 3.00 meters above the surface of
  the water in a pool. When she is 1.00 meter above
  the water, what are her gravitational potential
  energy and kinetic energy with respect to the
  water's surface?

In this situation, the force doing the work is
gravity, which is an internal force. KEi PEi
KEf PEf KEf 1080 J
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Example 2

• A spring in a toy car is compressed a distance,
  x. When released, the spring returns to its
  original length, transferring its energy to the
  car. Consequently, the car having mass m moves
  with speed v. Derive the spring constant, k, of
  the cars spring in terms of m, x, and v. Assume
  an ideal mechanical system with no loss of
  energy.

Since only force is elastic, TME is constant
KEi PEi KEf PEf
0 ½ kx2 ½ mv2 0 ½ kx2 ½ mv2 k mv2/x2
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Example 3

• The diagram shows a 0.1-kilogram apple attached
  to a branch of a tree 2 meters above a spring on
  the ground below. The apple falls and hits the
  spring, compressing it 0.1 meter from its rest
  position. If all of the gravitational potential
  energy of the apple on the tree is transferred to
  the spring when it is compressed, what is the
  spring constant of this spring?

Since only internal forces is doing work, TME is
constant
KEi PEi KEf PEf
0 mgh 0 ½ kx2 mgh ½ kx2 (0.1
kg)(9.81m/s2)(2m) ½k (0.1m)2 k 400 N/m
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Sample Problem 5E

• Starting from rest, a child zooms down a
  frictionless slide from an initial height of 3.00
  m. What is her speed at the bottom of the slide?
  Assume she has a mass of 25.0 kg.

7.67 m/s
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Class work

• Page 185 practice 5E

1. 20.7 m/s
2. 9.9 m/s 14.0 m/s
3. 14.1 m/s
4. 0.25 m
5. 0.18 m

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Mechanical Energy is not conserved when Wother ?
0
TMEi Wother TMEf

• When non-conservative forces do work
• TME changes,
• Wother TMEf TMEi
• When non-conservative forces is friction, heat is
  generated.

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Example 1
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Example 2
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Example 3
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Example 4

• A block weighing 15 N is pulled to the top of an
  incline that is 0.20 meter above the ground, as
  shown below. If 5.0 joules of work are needed to
  pull the block the full length of the incline,
  how much work is done against friction?

KEi PEi Wext KEf PEf The external forces
are applied force and friction force 0 0 Wapp
Wf 0 (15 N)(0.20 m) 5.0 J Wf 3.0J Wf
-2.0 J 2.0 J of work is done to overcome
friction
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Example 5

• In the diagram below, 450. joules of work is
  done raising a 72-newton weight a vertical
  distance of 5.0 meters. How much work is done to
  overcome friction as the weight is raised?

TMEi Wext TMEf
There are two external forces applied force and
friction force The applied force did 450 J of
work 0 450 J Wf (72 N)(5.0m) 450 J Wf
(72 N)(5.0 m) Wf -90 J 90 J of work is done to
overcome friction
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Example 6

• A box with a mass of 0.04 kg starts from rest at
  point A and travels 5.00 meters along a uniform
  track until coming to rest at point B, as shown
  in the picture. Determine the magnitude of the
  frictional force acting on the box. (assume the
  frictional force is constant.)

Given hA 0.80 m hB 0.50 m d 5.00 m m0.04
kg Unknown Ff ? N
TMEi Wother TMEf 0mg(0.80m)Wf0
mg(0.50m) Wf -0.12 J 0.12 J of work is done to
overcome friction Wf Ffd 0.12 J
Ff(5.00m) 0.12 J Ff 2.4 x 10-2 N
A
0.80 m
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Example 7

• A block weighing 40. newtons is released from
  rest on an incline 8.0 meters above the
  horizontal, as shown in the diagram below. If 50.
  joules of heat is generated as the block slides
  down the incline, what is the maximum kinetic
  energy of the block at the bottom of the incline?

KEi PEi Wext KEf PEf The external force
is friction force only 0 (40.N)(8.0m) Wf
KEf 0 320 J 50 J KEf KEf 270 J
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Example 8

• A person does 64 joules of work in pulling back
  the string of a bow. What will be the initial
  speed of a 0.5-kilogram arrow when it is fired
  from the bow?

16 m/s
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example

• Which of the following statements are true about
  work? Include all that apply.
• Work is a form of energy.
• Units of work would be equivalent to a Newton
  times a meter.
• A kgm2/s2 would be a unit of work.
• Work is a time-based quantity it is dependent
  upon how fast a force displaces an object.
• Superman applies a force on a truck to prevent it
  from moving down a hill. This is an example of
  work being done.
• An upward force is applied to a bucket as it is
  carried 20 m across the yard. This is an example
  of work being done.
• A force is applied by a chain to a roller coaster
  car to carry it up the hill of the first drop of
  the Shockwave ride. This is an example of work
  being done.

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example

• Determine the meaning of slope in each graph

elongation
K
1/K
A
B
force
Gravitational potential energy
mg
g
D
C
height
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Lab 17 - Energy of a Tossed Ball

• OBJECTIVES
• Measure the change in the kinetic and potential
  energies as a ball moves in free fall.
• See how the total energy of the ball changes
  during free fall.
• MATERIALS

computer ball
Vernier computer interface Logger Pro
Vernier Motion Detector wire basket
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PRELIMINARY QUESTIONS

• For each question, consider the free-fall portion
  of the motion of a ball tossed straight upward,
  starting just as the ball is released to just
  before it is caught. Assume that there is very
  little air resistance.
• 1. What form or forms of energy does the ball
  have while momentarily at rest at the top of the
  path?
• 2. What form or forms of energy does the ball
  have while in motion near the bottom of the path?
  
• 3. Sketch a graph of position vs. time for the
  ball.
• 4. Sketch a graph of velocity vs. time for the
  ball.
• 5. Sketch a graph of kinetic energy vs. time for
  the ball.
• 6. Sketch a graph of potential energy vs. time
  for the ball.
• 7. Sketch a graph of total energy vs. time for
  the ball.
• 8. If there are no frictional forces acting on
  the ball, how is the change in the balls
  potential energy related to the change in kinetic
  energy?

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DATA TABLE
Mass of the ball (kg)
Position Time Height Velocity PE KE TE
Position (s) (m) (m/s) (J) (J) (J)
After release
Top of path
Before catch
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ANALYSIS

1. Inspect kinetic energy vs. time graph for the
   toss of the ball.
2. Inspect potential energy vs. time graph for the
   free-fall flight of the ball.
3. Inspect Total energy vs. time graph for the
   free-fall flight of the ball.
4. Your conclusion from this lab
5. How does the kinetic and potential energy change?
6. How does the total energy change?

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Lab 17 - Energy of a Tossed Ball

• OBJECTIVES
• Measure the change in the kinetic and potential
  energies as a ball moves in free fall.
• See how the total energy of the ball changes
  during free fall.
• MATERIALS

computer ball
Vernier computer interface Logger Pro
Vernier Motion Detector wire basket
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PRELIMINARY QUESTIONS

• For each question, consider the free-fall portion
  of the motion of a ball tossed straight upward,
  starting just as the ball is released to just
  before it is caught. Assume that there is very
  little air resistance.
• 1. What form or forms of energy does the ball
  have while momentarily at rest at the top of the
  path?
• 2. What form or forms of energy does the ball
  have while in motion near the bottom of the path?
  
• 3. Sketch a graph of position vs. time for the
  ball.
• 4. Sketch a graph of velocity vs. time for the
  ball.
• 5. Sketch a graph of kinetic energy vs. time for
  the ball.
• 6. Sketch a graph of potential energy vs. time
  for the ball.
• 7. Sketch a graph of total energy vs. time for
  the ball.
• 8. If there are no frictional forces acting on
  the ball, how is the change in the balls
  potential energy related to the change in kinetic
  energy?

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DATA TABLE
Mass of the ball (kg)
Position Time Height Velocity PE KE TE
Position (s) (m) (m/s) (J) (J) (J)
After release
Top of path
Before catch
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ANALYSIS

1. Inspect kinetic energy vs. time graph for the
   toss of the ball.
2. Inspect potential energy vs. time graph for the
   free-fall flight of the ball.
3. Inspect Total energy vs. time graph for the
   free-fall flight of the ball.
4. Your conclusion from this lab
5. How does the kinetic and potential energy change?
6. How does the total energy change?