# 8-2 Potential Energy

```Announcement I
Physics 1408-002
Principles of Physics
Lecture 13
– Chapter 8 –
February 23, 2009
Sung-Won Lee
Sungwon.Lee@ttu.edu
Announcement II
SI session by
Reginald Tuvilla
SI sessions will be at the following times
and location.
Monday 4:30 - 6:00pm - Holden Hall 106
Thursday 4:00 - 5:30pm - Holden Hall 106
Lecture note is on the web
Handout (6 slides/page)
http://highenergy.phys.ttu.edu/~slee/1408/
*** Class attendance is strongly encouraged and will be
taken randomly. Also it will be used for extra credits.
HW Assignment #5 will be placed on
MateringPHYSICS today, and is due by
11:59pm on Wendseday, 2/25
Chapter 8
Conservation of
Energy
•! Conservative and Non-conservative Forces
•! Potential Energy
•! Mechanical Energy and Its Conservation
•! Problem Solving Using Conservation of Mechanical Energy
•! The Law of Conservation of Energy
•! Energy Conservation with Dissipative Forces: Solving Problems
•! Gravitational Potential Energy and Escape Velocity
•! Power
8-1 Conservative and Nonconservative Forces
A force is conservative if:
the work done by the force on an object
moving from one point to another depends
only on the initial and final positions of the
object, and is independent of the particular
path taken.
Example: gravity.
Object of mass m: (a) falls a height h vertically; (b) is raised along an
arbitrary two-dimensional path.
8-2 Potential Energy (P.E.)
•! We have now seen a method of storing energy in
a system - kinetic energy. The 2nd method, which we
cover in Chapter 8, is potential energy.
•! Potential energy is energy related to the configuration
of a system in which the components of the system
interact by forces.
•! Examples include:
–!elastic potential energy – stored in a spring
–!gravitational potential energy
–!electrical potential energy
8-2 Potential Energy
8-2 Potential Energy
In raising a mass m to a height
h, the work done by the
external force is
.
WG = FG d =
A person exerts an
upward force Fext = mg to
lift a brick from y1 to y2 .
mghcos1800
= -mgh
General definition of gravitational potential energy:
WG = -mgh
For any conservative force:
We therefore define the
gravitational potential
energy at a height y above
.
some reference point:
8-2 Potential Energy
A spring has potential
energy, called elastic
potential energy, when it is
compressed. The force
required to compress or
stretch a spring is:
8-2 Potential Energy
Then the potential energy is:
where k is called the
spring constant, and
needs to be measured for
each spring.
A spring (a) can store energy (elastic potential energy) when compressed
(b), which can be used to do work when released (c) and (d).
8-2 Potential Energy
In one dimension,
Conservation of Mechanical Energy
•! Look at work done by the book as it falls
from some height to a lower height
•! From work-kinetic energy theorem of Ch.7,
Won book = !Kbook
We can invert this equation to find U(x)
if we know F(x):
•! Also, W = F!r = mgyb – mgya = !Kbook
•! mgyb – mgya = -(Uf - Ui) = -!Ug
•! So, !K = -!Ug =&gt; !K + !Ug = 0 &quot;
In three dimensions:
•! We define the sum of kinetic and potential energies as
mechanical energy in the system: Emech = K + Ug
•! The statement of Conservation of Mechanical Energy for
an isolated system: Kf + Uf = Ki+ Ui
8-3 Mechanical Energy and Its
Conservation
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.
8-4 Problem Solving Using Conservation of
Mechanical Energy
In the image on the left,
the total mechanical
energy at any point is:
This allows us to define the total mechanical energy:
And its conservation:
.
8-4 Problem Solving Using Conservation of
Mechanical Energy
If the original height of the
rock is y1 = h = 3.0 m, calculate
the rock’s speed when it has
fallen to 1.0 m above the
ground.
8-4 Problem Solving Using Conservation of
Mechanical Energy
Assuming the height of the hill is 40 m, and the
roller-coaster car starts from rest at the top,
calculate (b) at what height it will have half this
speed. Take y = 0 at the bottom of the hill.
The rock’s potential energy changes to kinetic energy as it falls.
Note bar graphs representing potential energy U and kinetic
energy K for the three different positions.
8-4 Problem Solving Using Conservation of
Mechanical Energy
Assuming the height of the hill is 40 m, and the
roller-coaster car starts from rest at the top,
calculate (a) the speed of the roller-coaster car at
the bottom of the hill
8-4 Problem Solving Using Conservation of
Mechanical Energy
Estimate the kinetic energy and the speed
required for a 70-kg pole vaulter to just
pass over a bar 5.0 m high. Assume the
vaulter’s center of mass is initially 0.90 m
off the ground and reaches its maximum
height at the level of the bar itself.
8-4 Problem Solving Using Conservation of
Mechanical Energy
For an elastic force, conservation of energy tells us:
A dart of mass 0.100 kg is pressed
against the spring of a toy dart gun.
The spring (with spring constant k =
250 N/m and ignorable mass) is
compressed 6.0 cm and released. If
the dart detaches from the spring
when the spring reaches its natural
length (x = 0), what speed does the
dart acquire?
8-4 Problem Solving Using Conservation of
Mechanical Energy
Example 8-8: Two kinds of potential energy
A ball of mass m = 2.60 kg, starting
from rest, falls a vertical distance h
= 55.0 cm before striking a vertical
coiled spring, which it compresses
an amount Y = 15.0 cm.
8-4 Problem Solving Using Conservation of
Mechanical Energy
A dart of mass 0.100 kg is pressed against the
spring of a toy dart gun. The spring (with
spring constant k = 250 N/m and ignorable
mass) is compressed 6.0 cm and released. If the
dart detaches from the spring when the spring
reaches its natural length (x = 0), what speed
does the dart acquire?
A re-look at some problems
Let’s say that we want to know the velocity of a block sliding on
a frictionless inclined plane after it has slid down from a height h.
h
We determined the acceleration down
the plane before using
F = ma
s
!!
Determine the spring constant.
Assume the spring has negligible
mass, and ignore air resistance.
2
v2 = v2 + 2 a s = v + 2 g sin! (h / sin!)
0
0
v2 = v2 + 2 g h
0
v = &quot;2 g h
if it starts from rest
Using Energy Conservation
Let’s say that we want to know the velocity of a block sliding on
a frictionless inclined plane after it has slid down from a height h.
h
s
!!
Kf + Uf = Ki + Ui
2
1
– mv
2
f
+0=
1
–
2
Here the acceleration down the
plane is continually changing
since the angle of plane with
the horizontal changes.
h
mv2 + mgh
vf = &quot;2 g h
Kf + Uf = Ki + Ui
i
vf2 = vi 2 + 2 g h
if it starts from rest
Now what about a block sliding
down an incline like this
2
1
– mv
f
2
2
+ 0 =1– mvi 2 + mgh
2
2
vf = vi + 2 g h
if it starts from rest
vf = &quot;2 g h
A Swinging Pendulum
starts from rest at height h
Total Energy
is same everywhere.
solve for velocity
at bottom of swing
h
Kf + Uf = Ki + Ui
2
+ 0 =1– mv + mgh
2
1
– mv
f
2
2
i
2
2
v f = vi + 2 g h
v f= &quot;2 g h
8-5 The Law of Conservation of Energy
Nonconservative forces:
Friction
Heat
Electrical energy
Chemical energy
and more
do not conserve mechanical energy. However,
when these forces are taken into account, the
total energy is still conserved:
Work done by friction is always
negative
since Ffr is always opposite displacement
Wfr = – Ffr l = &micro;N l
where “l” is the total path length
(displacement)
Top of swing: v=0, K=0
(A,C)
all U
v = &quot;2 g h
Bottom of swing: U=0
(B) max v, All K
8-5 The Law of Conservation of Energy
The law of conservation of energy is one of the
most important principles in physics.
The total energy is neither increased nor
decreased in any process. Energy can be
transformed from one form to another, and
transferred from one object to another, but
the total amount remains constant.
When friction is acting,
mechanical energy is not conserved
#K = Wtot = \$ Wcons + \$ Wnon-cons
= – \$ ( #U) + \$ Wnon-cons
#K = – \$ ( #U) – Ffrl
Kf + \$ Uf = Ki + \$ Ui – Ffrl
Mechanical energy is not conserved due to friction force
Kf + \$ Uf &lt; Ki + \$ Ui
Where did it go ?
Block down a plane with friction
Let it start from rest
h
Kf + Uf = Ki + Ui – Ffrs
s
2
1
– mv
f
2
!!
2
1
– mv
f
2
= mgh – Ffrs
= mgh – (&micro; m g cos!) s
vf2 = 2 g h – 2 &micro;g cos! (h / sin! )
vf = &quot;2 g h (1 – &micro; cot !)
e.g. no friction force; vf = &quot;2 g h
8-6 Energy Conservation with Dissipative
Forces: Solving Problems
Determine the thermal energy produced and estimate
the average friction force (assume it is roughly
constant) on the car, whose mass is 1000 kg.
8-6 Energy Conservation with Dissipative
Forces: Solving Problems
Example: Friction on the roller-coaster car.
The roller-coaster car shown reaches a vertical height
of only 25 m on the second hill before coming to a
momentary stop. It traveled a total distance of 400 m.
Determine the thermal energy
produced and estimate the
average friction force (assume
it is roughly constant) on the
car, whose mass is 1000 kg.
8-6 Energy Conservation with Dissipative
Forces: Solving Problems
Example: Friction with a spring.
A block of mass m sliding along
a rough horizontal surface is
traveling at a speed v0 when it
strikes a massless spring headon and compresses the spring a
maximum distance X. If the
spring has stiffness constant k,
determine the coefficient of
kinetic friction between block
and surface.
Because of friction, a roller-coaster car does not reach the original height on the
second hill. The difference in the initial and final energies is the thermal energy
produced, 147000 J. This is equal to the average frictional force multiplied by the
distance traveled, so the average force is 370 N.
8-7 Gravitational Potential Energy and
Escape Velocity
Far from the surface of the Earth, the force of
gravity is not constant:
The work done on an object
moving in the Earth’s
gravitational field is given by:
Arbitrary path of particle of mass m moving
from point 1 to point 2.
8-7 Gravitational P.E. and Escape Velocity
Solving the integral gives:
We can define gravitational potential energy:
8-7 Gravitational Potential Energy and
Escape Velocity
Example: Package dropped from high-speed rocket.
A box of empty film canisters is allowed to fall from a
rocket traveling outward from Earth at a speed of
1800 m/s when 1600 km above the Earth’s surface.
The package eventually falls to the Earth. Estimate
its speed just before impact. Ignore air resistance.
8.8 Power
•! The time rate of energy transfer
•! The average power is given by
Instantaneous Power
•! The instantaneous power is the limiting value of
the average power as #t approaches zero
•! This can also be written as
Summary of Chapter 8
•!Gravitational potential energy: Ugrav = mgy.
•!Elastic potential energy: Uel = ! kx2.
•!For any conservative force:
•!Total mechanical energy is the sum of
kinetic and potential energies.
•!Additional types of energy are involved
when nonconservative forces act.
•!Gravitational potential energy:
•!Power:
8-7 Gravitational Potential Energy and
Escape Velocity
If an object’s initial kinetic energy is equal to the
potential energy at the Earth’s surface, its total
energy will be zero. The velocity at which this is
true is called the escape velocity; for Earth:
Power Generalized
•! Power can be related to any type of energy transfer
•! In general, power can be expressed as
Units of Power
•! The SI unit of power is called the [watt]
–! 1 watt = 1 joule / second = 1 kg . m2 / s2
•! US Customary system is horsepower: 1 hp = 746 W
•! Unit of energy can be defined in terms of units of
power.
–! 1 kWh (kilowatt-hour)= (1000 W)(3600 s) = 3.6 x106 J
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