# Work & Energy ```Chapter 6

So far, we’ve been obsessed with motion and
how it relates to Newton’s Laws and forces.

Now we will be concerned with two scalar
quantities, work &amp; energy and how they are
always conserved or remain constant.
ENERGY

WORK
the ability to do work

a transfer of energy
Wait, what?!
1.
Work Done by a Constant Force
2.
Work Done by a Varying Force
3.
Kinetic Energy, and the Work-Energy Principle
4.
Potential Energy
5.
Conservative and Nonconservative Forces
6.
Mechanical Energy and Its Conservation
7.
Problem Solving Using Conservation of Mechanical Energy
8.
Other Forms of Energy; Energy Transformations and the Law of
Conservation of Energy
9.
Energy Conservation with Dissipative Forces: Solving Problems
10.
Power
DEFINITION


IMPORTANT INFO
Scalar Quantity
a force acting through a
distance

the product of the
magnitudes of the
displacement times the
component of force
parallel to the
displacement.

Symbol: W

Units: Joule (J)
 Magnitude only
 1 J = 1 Nm

*W = Fd
 W – Work (J)
 F – Force (N)
 d – displacement (m)
- For
work
to be does
done:
• Recap
As long
as this
person
1.
force
must be
applied.
notAlift
or lower
the
bag of
2.
The object
must
move.
groceries,
he is
doing
no work
3.
on The
it. object must move in the
•


direction of the force.
The force he exerts has no
Iscomponent
the force ininthe
thedirection
directionofof
motion?
motion – it is perpendicular
 No – So, is work being done?
to the direction of motion.
What about a force acting at
an angle to the displacement?
The work done by a constant force is defined as
the distance the object moved multiplied by
the component of the force in the direction of
displacement:
θ is the angle
between the direction
of the F and d
1.
Draw a free-body diagram.
2. Choose a coordinate system.
3.
Apply Newton’s laws to determine any unknown
forces.
4. Find the work done by a specific force.
5.
To find the net work, either:
 find the net force and then find the work it does, or
 find the work done by each force and add.

A 50 kg crate is pulled 40 m along a horizontal floor
by a constant force exerted by a person, Fp = 100 N,
which acts at a 37&deg; angle as shown. The floor is
rough and exerts a friction force Ff = 50 N.
Determine the work done by each force acting on
the crate, and the net work done on the crate.
mg
1.
Draw a FBD.
FN
Fp
Ff
Fg
1.
Choose a coordinate system.
 Let’s choose +x to be in the direction of motion.
FN
Fp
d = 40 m
Ff
Fpx
Fg
Find any unknown forces using Newton’s 2nd.
3.

4.
All forces are known (or can be determined easily).
Determine the work done by each force.
FN
Fp = 100 N
θ = 37&deg;
Ff = 50 N
d = 40 m
Fg
Determine the work done by each force.
4.




WN = perpendicular to direction of motion = zero
Wg = perpendicular to direction of motion = zero
Wf = Fdcosθ = (50 N) (40 m) (cos 180) = -2000 J
WpII=Fdcosθ = (100 N) (40 m) (cos 37) = 3200 J
Determine the net work done on the crate.
5.

Wnet = 0 + 0 + (-2000 J) + (3200 J) = 1200 J
a)
Determine the work a hiker must
do on a 15.0 kg backpack to
carry it up a hill of height h = 10.0
m.
a)
Determine the work done by
gravity on the backpack.
a)
Determine the net work done on
the backpack. For simplicity,
assume the motion is smooth
and the velocity is constant.
Concept Questions
 p. 160 #3-6
Practice Problems
 p. 162 #2-5, 8 + 10
xkcd.com
What if the applied force is not constant - it keeps changing?
How can we determine work done by a varying force?



We could do my favorite thing!
▪ Let’s graph it! 
▪ Plot Fcosθ vs. d.
▪ What represents the work done on this type of graph?
The work done between any two points on the curve is equal
to the area under the curve between those two points.
Try #12 (a &amp; b) on p.162.

Kinetic Energy – the
energy of motion.

KE is measured in Joules.
Translational Kinetic Energy:

A moving object can do
work on another
moving object that it
strikes - it can exert a
force on an object and
move it a distance.
(Therefore, W is done.)
1
2
KE  ( mv )
2

KE is a scalar quantity
that is always either
zero or greater than
zero. Why?

The net work done on an
object is equal to the
change in the kinetic
energy of the object.
Wnet  KE
Wnet
1 2 1 2
 mv f  mvi
2
2
•
If the net work is
positive, the
kinetic energy
increases.
•
If the net work is
negative, the
kinetic energy
decreases.

Potential Energy is the
energy of position.

Examples of potential
energy:
• An object at some
height above the ground
• A stretched or
compressed spring
• A stretched elastic band
or string

In raising a mass m to a
height h, the work done by
the external force is

Therefore, we define the
gravitational potential
energy:
Ug = mgy

The energy an object
has because of its
position at some
height above the
ground.
Ug = mgh

Notice, the higher an
object is, the more
gravitational PE it has.


This potential energy
can become kinetic
energy if the object is
dropped.
Potential energy is a
property of a system as
a whole, not just of the
object (because it
depends on external
forces).

If Ug = mgy, where do
we measure y from?

It doesn’t matter, as
long as we are
we choose y = 0.

Only changes in
potential energy can be
measured.

Potential energy can also be stored in a spring
when it is compressed; the figure below
shows potential energy yielding kinetic
energy.

The force required to
compress or stretch a spring
is:
Where:

k is called the spring constant, and
needs to be measured for each
spring.

x is the displacement from the
unstretched position.

Also known as the spring equation
or Hooke’s Law.

The force increases as
the spring is stretched or
compressed.

We find that the
potential energy of the
compressed or
stretched spring,
measured from its
equilibrium position,
can be written:
Elastic PE, or Us = &frac12; kx2

Note: For a spring, we choose the
reference point for zero PE at the
spring’s natural position.
GRAVITATIONAL PE
ELASTIC PE
Concept Question
 p.161 #9
Concept Question
 p.161 #7
Practice Problems
 Review Example 6-7 on
p.146
Practice Problems
 Do #26, 29 + 32 on p.163

Do #27, 28, 30, 31 on p.163
CONSERVATIVE
NONCONSERVATIVE

Forces for which the work
done does not depend on
the path taken but only on
the initial and final
positions.

Forces for which the work
done depends not only on
the starting and ending
points but also on the path
taken.

Example: Gravity

Example: Friction

Although potential
energy is always
associated with a force
with a force, not all
forces have potential
energy.

Potential energy can
only be defined for
conservative forces.

Review Examples 6-4,
6-5, and 6-6 on page
143-144.
Complete the
Following Problems on
p.162:
 #15-20


Therefore, we must
distinguish between
the work done by
conservative forces
and the work done by
nonconservative
forces.

We find that the work
done by
nonconservative forces
is equal to the total
change in kinetic and
potential energies:
WNC = ΔKE + ΔPE

If there are no
nonconservative
forces, the sum of the
changes in the kinetic
energy and in the
potential energy is
zero – the kinetic and
potential energy
changes are equal but
opposite in sign.

This allows us to define
the total mechanical
energy (E):
E = KE + PE

And its conservation:
E1 = E2 = constant

In the image on the left,
the total mechanical
energy is:

If there is no friction, the speed of a roller
coaster will depend only on its height
compared to its starting height.

For an elastic force, conservation of energy
tells us:

Law of Conservation of
Energy: Energy cannot be
created nor destroyed. (It can
change form.)

Though there are several
types of energy, the total
energy of a closed system
remains constant.
 Changes are NOT perfect –

Energy is a scalar quantity
expressing the capacity to
do work.

There are many types of
energy. Most fit into two
1. Potential Energy
2. Kinetic Energy
When energy changes form,
some energy is lost as heat
(thermal energy).
 Friction also usually
accompanies these changes –
still lost as heat.

Some other forms of
energy:

Work is done when
energy is transferred
from one object to
another.

Accounting for all forms
of energy, we find that
the total energy neither
increases nor decreases.

Energy as a whole is
conserved.
 electric energy
 nuclear energy
 thermal energy
 chemical energy


If there is a
nonconservative force
such as friction, where
do the kinetic and
potential energies go?
They become heat; the
actual temperature
rise of the materials
involved can be
calculated.
Problem Solving: See p. 157
1.
Draw a picture.
2.
Determine the system for
which energy will be
conserved.
3.
Figure out what you are
looking for, and decide on the
initial and final positions.
4.
Choose a logical reference
frame.
5.
Apply conservation of energy.
6.
Solve.

Power is the rate at
which work is done.

In the SI system, the
units of power are
watts.
 1 W = 1 J/s = 1 N∙m/s
 1 horsepower = 746 W

So, the difference
between walking and
running up stairs is
power – the work done
and change in
gravitational potential
energy is the same.

Power is also needed for acceleration and for
moving against the force of gravity.

The average power can be written in terms of the
force and the average velocity:
•
Work:
•
Kinetic energy is energy of motion:
•
Potential energy is energy associated with forces that
depend on the position or configuration of objects.
•
The net work done on an object equals the change in its
kinetic energy – the work-energy principle.
•
If only conservative forces are acting, mechanical energy
is conserved.
•
Power is the rate at which work is done.
Conservation of Mechanical Energy
 p.163 #33, 34, 39, 42
Law of Conservation of Energy
 p.164 # 47 + 48
Power
 p.165 # 58, 59, 60
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