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Chapter 5
Energy
Forms of Energy

Mechanical




May be kinetic (associated with
motion) or potential (associated with
position)
Chemical
Electromagnetic
Nuclear
Work - Energy and Force




F is the magnitude
of the force
∆x is the
magnitude of the
object’s
displacement
q is the angle
between
Notes on Work

Gives no information about




Work is a scalar quantity
Work done by a force is zero when force
and displacement are perpendicular


time it took for the displacement to occur
the velocity or acceleration of the object
cos 90° = 0
For multiple forces, the total work done
is the algebraic sum of the amount of
work done by each force
More Notes on Work

SI

Newton • meter = Joule


US Customary

foot • pound

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N • m = J = kg • m2 / s2
ft • lb
Work can be positive or negative


Positive if the force and the displacement
are in the same direction
Negative if the force and the displacement
are in the opposite direction
Example of Sign for Work


Work is positive
when lifting the
box
Work would be
negative if
lowering the box

The force would
still be upward,
but the
displacement
would be
downward
Exmaple Problem

An eskimo pulls a sled
of salmon. A force of
120 N is exerted on the
sled via the rope to pull
the sled 5 m


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Find the work if q=0o
Find the work if q=30o
Does it seem odd that
less work is required in
the second case?!
Work and Dissipative
Forces


Work can be done by friction
The energy lost to friction by an object
goes into heating both the object and
its environment


Some energy may be converted into sound
For now, the phrase “Work done by
friction” will denote the effect of the
friction processes on mechanical energy
alone
Example 5.2, and a Lesson
in Graphical Display

Consider the eskimo
pulling the sled again.
The loaded sled has a
total mass of 50.0 kg


Find the net work done
for the previous two
cases
Consider the figure at
right for the normalized
net work as a function
of m and q
Kinetic Energy

Energy associated with the motion
of an object:
1
2
KE 


2
mv
Scalar quantity with the same
units as work
Work-Kinetic Energy Theorem


Speed will increase if work is positive
Speed will decrease if work is negative
Work and Kinetic Energy

An object’s kinetic
energy can be
likened to the work
that could be done if
object were brought
to rest (so, the K.E.
is like potential work
content)

The moving hammer
has kinetic energy
and can do work on
the nail
Example
Find the minimum
stopping distance for a
car traveling at 35.0 m/s
(about 80 mph) with a
mass of 1000 kg to avoid
backending the SUV.
Assume that braking is a
constant frictional force
of 8000 N.
Types of Forces

There are two general classes of
forces

Conservative

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Work and energy associated with the
force can be recovered
Nonconservative

The forces are generally dissipative and
work done against it cannot easily be
recovered
Friction Depends on Path

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The blue path is
shorter than the
red path
The work required
is less on the blue
path than on the
red path
Friction depends
on the path and
so is a nonconservative force
Potential Energy

Potential energy is associated with
the position of the object within
some system


Potential energy is a property of the
system, not the object
A system is a collection of objects
interacting via forces or processes
that are internal to the system
Work and Potential Energy


For every conservative force a
potential energy (PE) function can
be found
Evaluating the difference of the
function at any two points in an
object’s path gives the negative of
the work done by the force
between those two points:
W  (Pf  Pi )  P
Work and Gravitational
Potential Energy

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PE = mgy
Wgrav ity  PEi  PEf
Work-Energy Theorem,
Extended

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The work-energy theorem can be
extended to include potential energy:
If other conservative forces are present,
potential energy functions can be
developed for them and their change in
that potential energy added to the right
side of the equation
Conservation of Energy

Total mechanical energy is the
sum of the kinetic and potential
energies in the system
Ei  E f
KEi  PEi  KEf  PEf

Other types of potential energy
functions can be added to modify this
equation
Quick Quiz
Three balls are cast
from the same
point with the same
speed, but different
trajectories. Rank
their speeds (from
fast to slow) when
they hit the ground.
Example
A grasshopper
makes a leap as
shown at right, and
achieves a
maximum height of
1.00 m. What was
its initial speed vi?
Non-Conservative Forces


This young woman (at
m=60 kg) zips down a
waterslide and is
clocked at the bottom
at 18.0 m/s. If
conservative, she
should have been
moving at 20.7 m/s.
How much energy was
lost to friction, both as
an amount and as a
percentage?
Springs

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Involves the spring constant, k
Hooke’s Law gives the force

F=-kx
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F is the restoring force
F is in the opposite direction of x
k depends on how the spring was
formed, the material it is made from,
thickness of the wire, etc.
The force is conservative for “ideal”
springs, so there is an associated PE
function
Spring Potential Energy

Elastic Potential Energy


related to the work required to
compress a spring from its
equilibrium position to some final,
arbitrary, position x
1 2
PEs  kx
2
Work-Energy Theorem
Including a Spring

Wnc = (KEf – KEi) + (PEgf – PEgi) +
(PEsf – PEsi)

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PEg is the gravitational potential
energy
PEs is the elastic potential energy
associated with a spring
Classic Spring Problem
A block has mass m = 0.500 kg. The
spring has k = 625 N/m and is
compressed 10 cm.
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Find the distance d traveled if q= 30o.
How fast is the block moving at halfway up?
Nonconservative Forces
with Energy Considerations

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When nonconservative forces are
present, the total mechanical energy of
the system is not constant
The work done by all nonconservative
forces acting on parts of a system
equals the change in the mechanical
energy of the system

Nonconservative Forces
and Energy

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In equation form:
The energy can either cross a boundary
or the energy is transformed into a
form of non-mechanical energy such as
thermal energy (so the total energy is
still conserved, just not the sum of KE
and PE)
Power - Energy Transfer

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Often interested in the rate at which
energy transfer takes place
Power is defined as this rate of energy
transfer

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SI units are Watts (W, but not “Work”)
kg • m2
Center of Mass

The point in the body at which all
the mass may be considered to be
concentrated

When using mechanical energy, the
change in potential energy is related
to the change in height of the center
of mass
Work Done by Varying
Forces

The work done by
a variable force
acting on an
object that
undergoes a
displacement is
equal to the area
under the graph
of F versus x
Recall Spring Example

Spring is slowly
stretched from 0
to xmax

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W = 1/2 kx2
Spring Energy


The work is also
equal to the area
under the curve
In this case, the
“curve” is a triangle
Area = 1/2 X Base X height
gives W = 1/2 k x2