Work and Energy

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Work and Energy

• Unit 4

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Lesson 1 Work Done by a Constant Force
When a force acts on an object while displacement
occurs, the force has done work on the object.
The magnitude of work (W) is the product of the
amount of the force applied along the direction
of displacement and the magnitude of the
displacement.
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Units of Work
Determining the Sign of Work
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Example 1
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Example 2
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Graphical Analysis of Work
F
WF FDx
Dx
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Work is a Scalar (Dot) Product
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Example 3
a) Calculate the magnitudes of the displacement
and the force.
b) Calculate the work done by force F.
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Lesson 2 Work Done by a Varying Force
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As Dx approaches 0,
Therefore,
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Example 1
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Example 2
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This equation is in SI units, where x is the
Sun-probe separation distance. Determine how much
work is done by the Sun on the probe as the
probe-Sun separation changes from 1.5 x 1011 m to
2.3 x 1011 m.
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Graphical Solution
Each square (0.05 N)(0.1 x 1011 m) 5 x 108 J
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Work Done by a Spring
Negative sign signifies that the force exerted by
spring is always directed opposite to the
displacement.
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stretched spring
equilibrium position
compressed spring
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xf
ò
Ws
Fsdx
xi
½ kx2
Work done by the spring force is positive because
the force is in the same direction as
displacement.
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Generalized Work Done by Spring
Generalized Work Done on Spring
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Example 3
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a) If a spring is stretched 2.0 cm by a suspended
mass of 0.55 kg, what is the spring constant of
the spring ?
b) How much work is done by the spring as it
stretches through this distance ?
c) Suppose the measurement is made on an
elevator with an upward vertical acceleration
a. Will the unaware experimenter arrive at the
same value of the spring constant ?
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Example 4
If it takes 4.00 J of work to stretch a Hookes
Law spring 10.0 cm from its unstressed length,
determine the extra work required to stretch it
an additional 10.0 cm.
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Example 5
A light spring with spring constant 1200 N/m is
hung from an elevated support. From its lower end
a second light spring is hung, which has spring
constant 1800 N/m. An object of mass 1.50 kg is
hung at rest from the lower end of the second
spring.
a) Find the total extension distance of the pair
of springs.
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b) Find the effective spring constant of the
pair of springs as a system. We describe these
springs as in series.
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Example 6
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Lesson 3 Work-Kinetic Energy Theorem
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Kinetic Energy
Work - Kinetic Energy Theorem
If work done on a system only changes its speed,
the work done by the net force equals the change
in KE of the system.
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Example 1
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Example 2
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Example 3
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Lesson 4 Situations Involving Kinetic Friction
SFx max
(SFx)Dx (max)Dx
Dx ½ (vi vf) t
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(SFx)Dx ½ mvf2 ½ mvi2
(SFx)Dx -fkDx ½ mvf2 ½ mvi2 DKE
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DKE in General
OR
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Example 1
A 6.0 kg block initially at rest is pulled to the
right along a horizontal surface by a constant
horizontal force of 12 N.
a) Find the speed of the block after it has
moved 3.0 m if the surfaces in contact have a
coefficient of kinetic friction of 0.15.
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b) Suppose the force F is applied at and angle q
as shown below. At what angle should the force
be applied to achieve the largest possible speed
after the block has moved 3.0 m to the right ?
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Change in Internal Energy due to Friction
The result of a friction force is to transform KE
into internal energy, and the increase in
internal energy is equal is equal to the decrease
in KE.
DEsystem DKE DEint 0
-fkd DEint 0
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Example 2
A 40.0 kg box initially at rest is pushed 5.00 m
along a rough, horizontal floor with a constant
applied horizontal force of 130 N. If the
coefficient of friction between box and floor is
0.300, find
a) the work done by the applied force
b) the increase in internal energy in the
box-floor system due to friction
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c) the work done by the normal force
d) the work done by the gravitational force
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e) the change in kinetic energy of the box
f) the final speed of the box.
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Lesson 5 Power
Same amount of work done
Time interval is different
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Average Power
time rate of energy transfer
Instantaneous Power
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Units of Power
SI unit of power is J/s or the Watt (W).
In the U.S. customary system, the unit of power
is the horsepower (hp).
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The kilowatt-hour (kWh)
The energy transferred in 1 h at the constant
rate of 1kW 1000 J/s.
Note that a kWh is a unit of energy, not power.
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Example 1
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a) What power delivered by the motor is required
to lift the elevator car at a constant speed of
3.00 m/s ?
b) What power must the motor deliver at the
instant the speed of the elevator is v if the
motor is designed to provide the elevator car
with an upward acceleration of 1.00 m/s2 ?
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Example 2
Find the instantaneous power delivered by gravity
to a 4 kg mass 2 s after it has fallen from rest.
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Example 3
Find the instantaneous power delivered by the net
force at t 2 s to a 0.5 kg mass moving in one
dimension according to x(t) 1/3 t3.
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Example 4
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Example 5 AP 2003 1
a) Determine the speed of the box at time t 0.
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b) Determine the following as functions of time
t.
i. The kinetic energy of the box.
ii. The net force acting on the box.
iii. The power being delivered to the box.
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c) Calculate the net work done on the box in the
interval t 0 to t 2 s.
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Lesson 6 Potential Energy
Work done on system by external agent in lifting
book
When book is at yb, the energy of the system has
potential to become KE.
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Gravitational Potential Energy
Units for Ug are Joules (J). Like work and KE, Ug
is a scalar quantity.
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Example 1
A bowling ball held by a careless bowler slips
from the bowlers hands and drops on the bowlers
toe. Choosing floor level as the y 0 point of
your coordinate system, estimate the change in
gravitational PE of the ball-Earth system as the
ball falls. Repeat the calculation, using the top
of the bowlers head as the origin of coordinates.
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Example 2
A 400 N child is in a swing that is attached to
ropes 2.00 m long. Find the gravitational
potential energy of the child-Earth system
relative to the childs lowest position when
a) the ropes are horizontal
b) the ropes make a 30o angle with the vertical
c) the child is at the bottom of the circular
arc.
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Lesson 7 Conservation of Mechanical Energy
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Won book mgyb - mgya
DKEbook mgyb - mgya
For the book-Earth system,
mgyb mgya -(mgya mgyb) -(Uf Ui) -DUg
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Bringing DU to left side of the equation,
DKE DUg 0
(KEf KEi) (Uf Ui) 0
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Elastic Potential Energy
WFapp ½ kxf2 ½ kxi2
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Example 1
a) Neglecting air resistance, determine the
speed of the ball when it is at a height y
above the ground.
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b) Determine the speed of the ball at y if at
the instant of release it already has an
initial upward speed vi at the initial altitude
h.
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Example 2
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a) Find the speed of the sphere when it is at
the lowest point B.
b) What is the tension TB in the cord at B ?
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Example 3
a) Neglecting all resistive forces, determine
the spring constant.
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b) Find the speed of the projectile as it moves
through the equilibrium position of the spring
(where xB 0.120 m).
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Example 4
a) What is its speed at point A ?
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b) How large is the normal force on it if its
mass is 5.00 g ?
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Example 5
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Example 6 AP 1989 1
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a) Determine the height h.
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c) Determine the magnitude of the force exerted
by the track on the block when it is at point B.
d) Determine the maximum height above the ground
attained by the block after it leaves the track.
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e) Another track that has the same
configuration, but is NOT frictionless, is
used. With this track it is found that if the
block is to reach point C with a speed of 4 m/s,
the height h must be 2 m. Determine the work
done by the frictional force.
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Example 7 AP 1985 2
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In terms of the given quantities, derive an
expression for each of the following.
a) ms, the coefficient of static friction
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b) DE, the loss in total mechanical energy of
the block-spring system from the start of the
block down the incline to the moment at which it
comes to rest on the compressed spring
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c) mk, the coefficient of kinetic friction
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Lesson 8 Conservative and Nonconservative Forces
Conservative Forces
1. The work done by a conservative force on a
particle moving between any two points is
independent of the path taken by the particle.
2. The work done by a conservative force on a
particle moving through any closed path is zero.
(A closed path is one in which the beginning and
end points are identical.)
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Examples of Conservative Forces
a) Gravitational Force
Wg mgyi - mgyf
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b) Force exerted by a spring
Ws ½ kxi2 ½ kxf2
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Nonconservative Forces
A force that does not satisfy the properties of a
conservative force.
Work done by force depends on the path.
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If book is displaced along blue path, work done
against friction is less than if book is pushed
along curved brown path.
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If the forces acting on objects within a system
are conservative, then the mechanical energy of
the system is conserved.
If some of the forces acting on objects within a
system are nonconservative, then the mechanical
energy of the system changes.
If a friction force acts within a system,
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Example 1
A 3.00 kg crate slides down a ramp. The ramp is
1.00 m in length and inclined at an angle of
30.0o. The crate starts from rest at the top,
experiences a constant friction force of
magnitude 5.00 N, and continues to move a short
distance on the horizontal floor after it leaves
the ramp. Use energy methods to determine the
speed of the crate at the bottom of the ramp.
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Diagram for Example 1
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Example 2
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a) Determine his speed at the bottom, assuming
no friction is present.
b) If a force of kinetic friction acts on the
child, how much mechanical energy does the
system lose ? Assume that vf 3.00 m/s and m
20.0 kg.
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Example 3
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Lesson 9 Conservative Forces and PE
The work done by a conservative force equals the
decrease in PE of the system.
DU is negative when Fx and dx are in the same
direction.
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F
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dU -Fx dx
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Elastic PE
Gravitational PE
Fg -mg
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Example 1
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Lesson 10 Energy Diagrams
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The force at a given point is the negative slope
of the curve.
Where the graph reaches maxima or minima, the
force will be 0.
Stable equilibrium points will be located at the
minima.
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Fx is positive
Fx is negative
Acceleration away from x 0
Acceleration away from x 0
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Example 1
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a) determine whether the force Fx is positive,
negative, or zero at the five points indicated.
b) indicate points of stable, unstable, and
neutral equilibrium.
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c) sketch the curve for Fx vs. x from x 0 to x
9.5 m
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Example 2
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a) Identify each equilibrium position for this
particle. Indicate whether each is a point of
stable, unstable, or neutral equilibrium.
b) The particle will be bound if the total
energy of the system is in what range ?
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Now suppose that the system has energy -3J.
Determine
c) the range of positions where the particle can
be found.
d) its maximum kinetic energy.
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e) the location where it has maximum kinetic
energy.
f) the binding energy of the system that is,
the additional energy that it would have to be
given in order for the particle to move out to r
? infinity .
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Example 3
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Taking D 50.0 m, F 110 N, L 40.0 m, and q
50.0o,
a) with what minimum speed must Jane begin her
swing in order to just make it to the other side
?
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b) Once the rescue is complete, Tarzan and Jane
must swing back across the river. With what
minimum speed must they begin their swing ?
Assume that Tarzan has a mass of 80.0 kg.
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Example 4 AP 1987 2
a) Identify all points of equilibrium for this
particle.
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Suppose the particle has a constant total energy
of 4.0 J, as shown by the dashed line on the
graph.
ii. x 4.0 m
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c) Can the particle reach the position x 0.5 m
? Explain.
d) Can the particle reach the position x 5.0 m
? Explain.
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e) On the grid below, carefully draw a graph of
the conservative force acting on the particle as
a function of x, for 0ltxlt7 m.
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Example 5 AP 1995 2
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a) In terms of the constants a and b, determine
the following
i. The position ro at which the potential energy
is a minimum.
ii. The minimum potential energy Uo.
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b) Sketch the net force on the particle as a
function of r on the graph below, considering a
force directed away from the origin to be
positive, and a force directed toward the origin
to be negative.
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The particle is released from rest at r ro/2.
c) In terms of Uo and m, determine the speed of
the particle when it is at r ro.
d) Write the equation or equations that could be
used to determine where, if ever, the particle
will again come to rest. It is not necessary to
solve for this position.
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e) Briefly and qualitatively describe the motion
of the particle over a long period of time.