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Section 5-3: Conservation of energy
Objectives
1. Identify situations in which conservation of
mechanical energy is valid.
2. Recognize the forms that conserved energy
can take.
3. Solve problems using conservation of
mechanical energy.
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
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.
Conservative vs. non - conservative Forces
• There are a variety of ways to categorize all the types
of forces.
1. Contact force: Forces that arise from the physical
contact of two objects.
2. 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.
1. Conservative force can never change the total
mechanical energy of an object
2. Non-conservative forces will change the total
mechanical energy of the object
• The conservative forces include the gravity forces,
spring force, magnetic force, electrical force.
• We will simply say all the other forces are nonconservative 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
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
PEgi + PEsi + KEi + Wother = PEgf + PEsf + KEf
TMEi + Wother = TMEf
Wother = TMEf - TMEi
• When net work is done upon an object by an nonconservative 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.
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.
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
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.
• 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.
0.306
0.153
1.73
0
2.45
0.306
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.
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.
A free falling object
• If a stationary object having mass m is located a vertical
distance h above Earth’s 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.
Energy conversion of a free falling object
The graph shows as a ball is dropped, how its energy is
transformed.
constant
• The total mechanical energy remains _____________.
• GPE decreases as KE increases
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
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 car’s
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
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) = ½∙k (0.1m)2
k = 400 N/m
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
Class work
• Page 185 – practice 5E
1.
2.
3.
4.
5.
20.7 m/s
9.9 m/s; 14.0 m/s
14.1 m/s
0.25 m
0.18 m
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.
Example 1
Example 2
Example 3
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
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
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.)
A
Given:
hA = 0.80 m
hB= 0.50 m
d = 5.00 m
m=0.04 kg
0.80 m
Unknown:
Ff = ? N
TMEi + Wother = TMEf
0+mg(0.80m)+Wf=0+ mg(0.50m)
Wf= -0.12 J
0.12 J of work is done to
overcome friction
B
Wf = Ff∙d = 0.12 J
Ff∙(5.00m) = 0.12 J
0.50 m Ff = 2.4 x 10-2 N
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
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
example
• Which of the following statements are true about work? Include
all that apply.
1. Work is a form of energy.
2. Units of work would be equivalent to a Newton times a meter.
3. A kg•m2/s2 would be a unit of work.
4. Work is a time-based quantity; it is dependent upon how fast a
force displaces an object.
5. Superman applies a force on a truck to prevent it from moving
down a hill. This is an example of work being done.
6. 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.
7. 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.
example
K
A
B
elongation
force
• Determine the meaning of slope in each graph
force
weight
Gravitational
potential
energy
elongation
g
C
1/K
mg
D
mass
height
Lab 17 - Energy of a Tossed Ball
OBJECTIVES
1. Measure the change in the kinetic and potential energies
as a ball moves in free fall.
2. See how the total energy of the ball changes during free
fall.
MATERIALS
computer
Vernier computer interface
Vernier Motion Detector
ball
Logger Pro
wire basket
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 ball’s potential energy related to the change in
kinetic energy?
DATA TABLE
Mass of the ball (kg)
Position
After
release
Top of
path
Before
catch
Time
(s)
Height
(m)
Velocity
(m/s)
PE
(J)
KE
(J)
TE
(J)
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
1. How does the kinetic and potential energy
change?
2. How does the total energy change?
Lab 17 - Energy of a Tossed Ball
OBJECTIVES
1. Measure the change in the kinetic and potential energies
as a ball moves in free fall.
2. See how the total energy of the ball changes during free
fall.
MATERIALS
computer
Vernier computer interface
Vernier Motion Detector
ball
Logger Pro
wire basket
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 ball’s potential energy related to the change in
kinetic energy?
DATA TABLE
Mass of the ball (kg)
Position
After
release
Top of
path
Before
catch
Time
(s)
Height
(m)
Velocity
(m/s)
PE
(J)
KE
(J)
TE
(J)
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
1. How does the kinetic and potential energy
change?
2. How does the total energy change?