Introduction to Work
Energy and Work
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A body experiences a change in energy when one
or more forces do work on it. A body must move
under the influence of a force or forces to say work
was done.
A force does positive work on a body when the
force and the displacement are at least partially
aligned. Maximum positive work is done when a
force and a displacement are in exactly the same
direction.
If a force causes no displacement, it does zero
work.
Forces can do negative work if they are pointed
opposite the direction of the displacement.
Calculating Work in Physics B
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If a force on an object is at least partially aligned
with the displacement of the object, positive work is
done by the force. The amount of work done
depends on the magnitude of the force, the
magnitude of the displacement, and the degree of
alignment.
W= F r cos q
F
F
q
q
r
Forces can do positive or negative work.

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When the load goes
up, gravity does
negative work and
the crane does
positive work.
F
When the load goes
down, gravity does
positive work and the
crane does negative
work.
mg
Units of Work
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SI System:

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British System:
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Joule (N m)
foot-pound
Atomic Level:

electron-Volt (eV)
Work and a Pulley System
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A pulley system, which has at least
one pulley attached to the load, can
be used to reduce the force
necessary to lift a load.
Amount of work done in lifting the
load is not changed.
The distance the force is applied over
is increased, thus the force is
reduced, since W = Fd.
F
m
Work as a “Dot Product”
Calculating Work a Different Way
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Work is a scalar resulting from the multiplication of
two vectors.
We say work is the “dot product” of force and
displacement.
W=F•r
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W= F r cos q
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dot product representation
useful if given magnitudes and directions of vectors
W = Fxrx + Fyry + Fzrz

useful if given unit vectors
The “scalar product” of two vectors is
called the “dot product”
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The “dot product” is one way to multiply two
vectors. (The other way is called the “cross
product”.)
Applications of the dot product
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Work
Power
Magnetic Flux
W=Fd
P=Fv
ΦB = B  A
The quantities shown above are biggest
when the vectors are completely aligned and
there is a zero angle between them.
Why is work a dot product?
F
q
s
W=F•r
W = F r cos q
Only the component of force aligned
with displacement does work.
Work by Variable Forces
Work and Variable Forces
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For constant forces
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For variable forces, you can’t move far until
the force changes. The force is only
constant over an infinitesimal displacement.
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W=F•r
dW = F • dr
To calculate work for a larger displacement,
you have to take an integral

W =  dW =  F • dr
Work and variable force
The area under the
curve of a graph of
force vs
displacement gives
the work done by
the force.
F(x)
xb
W = x F(x) dx
a
xa
xb
x
• Problem: Determine the work done by the force as the
particle moves from x = 2 m to x = 8 m.
F (N)
40
20
0
-20
-40
2
4
6
8
10
12
x (m)
Work Energy Theorem
Net Work or Total Work
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An object can be subject to many forces at the same
time, and if the object is moving, the work done by
each force can be individually determined.
At the same time one force does positive work on
the object, another force may be doing negative
work, and yet another force may be doing no work
at all.
The net work, or total, work done on the object (Wnet
or Wtot) is the scalar sum of the work done on an
object by all forces acting upon the object.
Wnet = ΣWi
The Work-Energy Theorem
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Wnet = ΔK
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When net work due to all forces acting upon an
object is positive, the kinetic energy of the object
will increase.
When net work due to all forces acting upon an
object is negative, the kinetic energy of the object
will decrease.
When there is no net work acting upon an object,
the kinetic energy of the object will be unchanged.
(Note this says nothing about the kinetic energy.)
Kinetic Energy
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Kinetic energy is one form of mechanical energy,
which is energy we can easily see and
characterize. Kinetic energy is due to the motion
of an object.
K = ½ m v2
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K: Kinetic Energy in Joules.
m: mass in kg
v: speed in m/s
In vector form, K = ½ m v•v
Power
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Power is the rate of which work is done.
No matter how fast we get up the stairs, our
work is the same.
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When we run upstairs, power demands on our
body are high.
When we walk upstairs, power demands on our
body are lower.
Pave = W / t
Pinst = dW/dt
P=F•v
Units of Power
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Watt = J/s
ft lb / s
horsepower
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550 ft lb / s
746 Watts
How We Buy Energy…
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The kilowatt-hour is a commonly used unit by
the electrical power company.
Power companies charge you by the kilowatthour (kWh), but this not power, it is really
energy consumed.
Conservative and NonConservative Forces
More about force types
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Conservative forces:
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Work in moving an object is path independent.
Work in moving an object along a closed path is zero.
Work is directly related to a negative change in potential
energy
Ex: gravity, electrostatic, magnetostatic, springs
Non-conservative forces:
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Work is path dependent.
Work along a closed path is NOT zero.
Work may be related to a change in mechanical energy, or
thermal energy
Ex: friction, drag, magnetodynamic
Potential Energy
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A type of mechanical energy possessed by an
object by virtue of its position or configuration.
Represented by the letter U.
Examples:
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Gravitational potential energy, Ug.
Electrical potential energy , Ue.
Spring potential energy , Us.
The work done by conservative forces is the
negative of the potential energy change.

W = -ΔU
Gravitational Potential Energy (Ug)
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The change in gravitational potential
energy is the negative of the work done by
gravitational force on an object when it is
moved.
For objects near the earth’s surface, the
gravitational pull of the earth is roughly
constant, so the force necessary to lift an
object at constant velocity is equal to the
weight, so we can say
ΔUg = -Wg = mgh
Note that this means we have defined the
point at which Ug = 0, which we can do
arbitrarily in any given problem close to
the earth’s surface.
h
Fapp
mg
Spring Potential Energy, Us
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Springs obey Hooke’s Law.
Fs(x) = -kx
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Ws =  Fs(x)dx = -k  xdx
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Fs is restoring force exerted BY the spring.
Ws is the work done BY the spring.
U s = ½ k x2
Unlike gravitational potential energy, we
know where the zero potential energy point is
for a spring.
Conservation of Mechanical
Energy
System
Boundary
Law of Conservation of Energy
The system is isolated and boundary
allows no exchange with the
environment.
E = U + K + Eint
= Constant
No mass can enter or leave!
No energy can enter or leave!
Energy is constant, or conserved!
Law of Conservation of
Mechanical Energy
We only allow U
and K to
interchange.
We ignore Eint
(thermal energy)
E=U+K
= Constant
Law of Conservation of Mechanical
Energy
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E=U+K=C
for gravity
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or
E = U + K = 0
 Ug = mghf - mghi
 K = ½ mvf2 - ½ mvi2
Assuming acceleration is constant
for springs
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 Us = ½ kxf2 - ½ kxi2
 K = ½ mvf2 - ½ mvi2
Assuming “hookean” spring
Pendulum Energy
h
½mv12 + mgh1 = ½mv22 + mgh2
For any points two points in the pendulum’s swing
Spring Energy
0
m
½ kx12 + ½ mv12
= ½ kx22 + ½ mv22
-x
For any two points in a spring’s
oscillation
m
m
x
Non-conservative Forces and
Conservation of Energy
Non-conservative forces
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Non-conservative forces change the
mechanical energy of a system.
Examples: friction and drag
Wtot = Wnc + Wc = K
Wnc = K – Wc
Wnc = K + U
Force and Potential Energy
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In order to discuss the relationships between
potential energy and force, we need to review
a couple of relationships.
Wc = Fx (if force is constant)
Wc =  Fdx = - dU = -U (if force varies)
 Fdx = - dU
Fdx = -dU
F = -dU/dx
Remember
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F = -dU/dx
F = dK/dx
W =  Fdx