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Work and Energy
Today’s Agenda



Work and Energy
 Definition of work
 Examples
Definition of Mechanical Energy
Conservation of Mechanical Energy
 Conservative forces
Conservation of Mechanical Energy, Pg 1
Give it a try:

What work does gravity do as my book is moved
upwards?
(a) W = mgy
(b) W = 0
(c) W = -mgy
final
mg
y
initial
mg
Conservation of Mechanical Energy, Pg 2
Work done by gravity

Lets compute the work done by the gravitational
force.
force
final
mg
y
initial
mg


W = F Cos   d
W = mg Cos 180  h = -mgy
Conservation of Mechanical Energy, Pg 3
Give it a try:
A crane lowers a girder into place at constant speed.
Consider the work Wg done by gravity and the work
WT done
d
b
by th
the ttension
i iin th
the cable.
bl Whi
Which
h iis ttrue?
?
A.
Wg > 0 and WT > 0
B.
Wg > 0 and WT < 0
C.
Wg < 0 and WT > 0
D.
Wg < 0 and WT < 0
E.
Wg = 0 and WT = 0
Conservation of Mechanical Energy, Pg 4
Conservative forces




For conservative forces the work done does not depend on
path taken,, only
p
y the starting
g and finishing
gp
points matter.
 Ex. gravity
For conservative force work on a closed path is zero
 Ex. Roller coaster
This semester
 Conservative
C
ti forces
f
 Gravity
 Springs
 Non-conservative forces
y
g else!
 Anything
Conservative forces give objects a potential energy!!!!
Conservation of Mechanical Energy, Pg 5
Potential energy

Potential energy: Stored energy
Energy depends on the position or configuration
of an object.

Potential energy due to gravity

Potential energy of a spring.

U = -Wconservative

Conservation of Mechanical Energy, Pg 6
Work done by gravity

Lets compute the work done by the gravitational
force.
force
yf
final
mg
yi
initial
mg


By moving the
block up
pa
distance y, it
gains potential
energy of mgy!
U = -WC
U = -(-mg(yf - yi)) = mg(y)
Conservation of Mechanical Energy, Pg 7
Gravitational potential energy

Potential energy due to earth’s
earth s gravitational field
 Energy of Position
 Work
o by g
gravity
a y W = -mg(y)
g( y)
 Work is energy given to object!
For an object at height y
and mass m.
Ug = mgy + Ug0

y
Usually we call U0 = 0
Ug = mgy
Conservation of Mechanical Energy, Pg 8
The WorkWork-Energy Theorem

Wtot = K

WC + WNC = K

WNC = K – WC

WNC = K + U
Conservation of Mechanical Energy, Pg 9
Mechanical energy

The total mechanical energy is defined as the
sum of the potential energy and the kinetic
energy for an object.
Emech = K + U
WNC = K + U = (Kf - Ki) + (Uf - Ui)
WNC = ((Kf + Uf) - ((Ki + Ui)
WNC = Ef - Ei
Conservation of Mechanical Energy, Pg 10
Conservation of Mechanical Energy

So if only conservative forces are doing work ((ie
ie
WNC is zero)
zero), the total mechanical energy of a
system is conserved.
WNC = Ef - Ei = 0
Einitial
i iti l = Efinal
fi l
E = K + U is constant
constant!!!
!!!

Both K and U can change, but E = K + U
remains constant.
constant
Conservation of Mechanical Energy, Pg 11
Work by a Non
Non--constant Force

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
Physics 201: Lecture 13, Pg 12
Work by a Non
Non--constant Force

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
F
x1
-kx1
W = ½ (-kx1)(x1)
W = -½
½ k(x1)2
F = -kx
x
WC = -U
U = ½ kx2
Physics 201: Lecture 13, Pg 13
Clicker Question 4:

You grasp the end of a spring that is attached to the wall
and is initially in its resting position
position. You pull it out until it
is extended 0.1 m from its resting position, then push it in
until it is compressed by 0.1 m from its resting position.
Finally you return the spring to its resting position
Finally,
position. The
spring constant is k = 20 N/m. The total work W done by
the spring on your hand is
(a) W < 0
(b) W = 0
(c) W > 0
Physics 131: Lecture 15, Pg 14
New Conservation of Mechanical Energy
G d when
Good
h only
l gravity
it or a spring-like
i
lik force
f
are doing
d i
work on an object
Ef = Ei
Kf + Uf = Ki + Ui
½ksf2 + ½mv
½k
½ f2 + mgh
hf = ½ks
½k i2 + ½mv
½ i2 + mgh
hi
Physics 201: Lecture 13, Pg 15
DEFINITION OF AVERAGE POWER



Average power is the rate at which work is done
done, and it
is obtained by dividing the work by the time required to
perform the work.
PAVG
E

t
Joule s  Watt (W)
Physics 201: Lecture 13, Pg 16
Give it a try:
Four students run up the stairs in the time shown
shown.
Which student has the largest power output?
Physics 201: Lecture 13, Pg 17