Potential Energy and Energy Conservation
Warm-Up: The Flying (and Driving) Dutchman
Stuck in traffic? Can’t make to be in time in 9:00am Phys250 class? What about the ability to fly in your own car?
o Dutch design engineering firm has just developed a three-wheeled vehicle that travels both on ground and in air, via a set of unfolding helicopter blades.
o The PALV (personal air and land vehicle), powered by a rotary engine, has a top ground speed of 125 mph (120 mph in the air) and can get between 60 and 70 miles per gallon of conventional gasoline. It can take off at close range, and can land vertically.
o We will se how this project will be developed…
o What is the thrust (forward force on the PALV) developed by the PALV Rotary engine with power output 213 hp when the vehicle is airborne and traveling in air horizontally at 120 mph?
http://www.sparkdesign.nl
Warm-Up: Power
Power climb Runner with mass m runs up the stairs to the top of 443-m-tall Sears
Tower. To lift herself there in 15 minutes (900 s), what must be her average power output in watts? Kilowatts? Horsepower?
JmkgmghWsm 51017.2)443)(8.9)(50( 2
Treat the runner as a particle of mass m. Let’s find first how much work she must do
against the gravity to lift herself at height h.
hpkWWs
J
t
WPav 323.0241.0241
900
1017.2 5
WkgvmgvFP sm
sm
avavav 241)492.0)(8.9)(50()( 2
Another way: calculate average upward force and then multiply by upward velocity
Upward force here is vertical, average vertical component of velocity is (443m) / (900s) = 0.492 m/s
Gravitational Potential Energy
Gravitational Potential Energy
Gravitational Potential Energy
Energy associated with position is called potential energy If elevation for which the gravitational potential energy is chosen to be
zero has been selected then the expression for the gravitational potential energy as a function of position y is given by
Gravitational potential energy Ugrav is associated with the work done by the gravitational force according to
mgyU grav
UUUUUWgrav )( 1221
Conservative with Non-Conservative Forces
Conservative and Non-Conservative Forces
Work done by the conservative force only depends on the initial and final positions, and doesn’t depend on the path
Runner: gravitational force is conservative From point 1 to point 2, same work
The work done by a conservative force has these properties:
It can always be expressed as the difference between the initial and final values of a potential energy function: U = -W.
It is reversible.
It is independent of the path of the body and depends only on the starting and ending points.
When the initial and final points are the same (closed loop), the total work is zero.
All forces which do not satisfy these properties are non-conservative forces.
Warm-Up: Gravitational Potential Energy
When this guy is in midair, only gravity does work on him (air resistance can be
neglected)
Mechanical energy (sum of kinetic and gravitational
potential energy) is conserved
E = K + U = const
Warm-Up: Gravitational Potential Energy
W = m g
Moving upK decreasesU increases
Moving downK increasesU decreases
When this guy is in midair, only gravity does work on him (air resistance can be
neglected)
Mechanical energy (sum of kinetic and gravitational
potential energy) is conserved
E = K + U = const
Warm-Up: Work due to Gravity
dx x
m mG - dx F =W
f
i
f
i
x
x2
21
x
x
if21
21
x
1 -
x
1 m mG - =
x
1- m mG - =W
f
i
x
x
x
m mG -= F
221 mg= F
dx mg - dx F =W f
i
f
i
x
x
x
x
)x-(x mg - =mg- =W iff
i
x
x
Near the Earth Away from the Earth
Warm-Up: Extinction
They disappeared at the boundary between the Cretaceous and Tertiary periods (C-T boundary)
70 Million years ago
Dinosaurs ruled the Earth
Luis Alvarez (1911 – 1988) ~ Nobel Prize winner in Physics ~ suggested an asteroid impact might be responsible
Why ?
Warm-Up: Extinction
Alvarez calculated the asteroid would need to be 10 km across and would leave a crater 150 km in diameter
A crater off the Yucatan peninsula of Mexico has been identified as a possible impact site. Research on this crater has shown it is the result of a extra-terrestrial impact.
Warm-Up: Extinction
Many asteroids and comets that cross the Earth’s path originate in the Oort cloud.
Most asteroids that hit the Earth originate in the inner Oort cloud that extends from 40 to 10,000 times the radius of the earth’s orbit from the sun.
This is a dense ring of asteroids that surrounds our solar system
Warm-Up: Extinction
J 10 x 8.9 = m 10 x 7.5
1 -
m 1.5x10
1
kg) kg)(10 10)(1.99 10 (6.672 =
x
1 -
x
1 m mG =W
241411
1630
kgmN11-
ifas
2
2
sm
16
24
42,100 =kg 10
J) 10 (8.9 2 =
m
K 2 = v
1 Ton TNT = 4109 J
Asteroid Impact:
2x109 MT TNT
Over 80,000 MPH !
Assume an asteroid started at rest in the middle of the inner Oort cloud (~5000 RE-S)
Assume it is acted on primarily by the Sun Assume mass ~1016 kg (10-km-rock)
Energy of the impact
Quick Reminder: 30º-60º-90º Triangle
12
3
30º
60º
90sin60sin30sin
BCACAB
CA
B
Elastic Potential Energy
Elastic Potential Energy
When you compress a spring: If there is no friction, spring moves back Kinetic energy has been “stored” in the
elastic deformation of the spring
Rubber-band slingshot: the same principle Work is done on the rubber band by the
force that stretches it That work is stored in the rubber band
until you let it go You let it go, the rubber gives kinetic
energy to the projectile
Elastic body: if it returns to its original shape and size after being deformed
Elastic Potential Energy
EquilibriumSpring is stretchedIt does negative work on block
Spring relaxes It does positive work on block
Spring is compressedPositive work on block
Block moves from one position x1 to another position x2: how much work does the elastic (spring) force do on the block?
Elastic Potential Energy
Work done ON a spring to move one end from elongation x1 to a different elongation x2
When we stretch the spring, we do positive work on the spring
When we relax the spring, work done on the spring is negative
Work done BY the spring From N3L: quantities of work are
negatives of each other
Thus, work Wel done by the spring
We can express the work done BY the spring in terms of a given quantity at the beginning and end of the displacement
JkxU 2
2
1
21
22 2
1
2
1kxkxW
22
21 2
1
2
1kxkxWel
Elastic potential energy
Elastic Potential Energy
2
2
1kxU
The graph of elastic potential energy for ideal spring is a parabola
For extension of spring, x>0 For compression, x<0 Elastic potential energy U is NEVER
negative! In terms of the change of potential
energy:
22
21
21
2
1
2
1kxkx
UUUWel
Elastic Potential Energy
When a stretched spring is stretched greater, Wel is negative and U increases: greater amount of elastic potential energy is stored in the spring
When a stretched spring relaxes, x decreases, Wel is positive and U decreases: spring loses its elastic potential energy
More spring compressed OR stretched, greater its elastic potential energy
22
2121 2
1
2
1kxkxUUUWel
Elastic Potential Energy: Work - Energy Theorem
12 UUWW eltot Work – Energy Theorem: Wtot=K2-K1, no matter
what kind of forces are acting on the body. Thus:
22111221 UKUKKKUUWtot If only elastic force does work
22
22
21
21 2
1
2
1
2
1
2
1kxmvkxmv
Total mechanical energy E (the sum of elastic potential energy and kinetic energy) is conserved
Ideal spring is frictionless and massless If spring has a mass, it also has kinetic energy Your car has a mass of 1.2 ton or more Suspension spring has a mass of few kg So we can neglect spring’s mass in study of how the car
bounces on its suspension
UKE
Elastic Force + other forces?
12 KKWWW othereltot If forces other than elastic force also do work on the body, the total work is
2211 UKWUK other elastic force + other forces
22
22
21
21 2
1
2
1
2
1
2
1kxmvWkxmv other
The work done by all forces other than the elastic force equals the change in the total mechanical energy E of the system, where U is the elastic potential energy:
“System” is made up of the body of mass m and the spring of force constant k
When Wother is positive, E increases
When Wother is negative, E decreases
UKE
Elastic Potential Energy: Example
Both Gravitational Potential Energy and Elastic Potential Energy
Spring with a body is hanged vertically Bungee jumper
U1 and U2 then are initial and final values of the total potential energy
1,1,1 elgrav UUU
2,2,21,1,1 elgravotherelgrav UUKWUUK
2,2,2 elgrav UUU
The work done by all forces other than the gravitational force or elastic force equals the change in the total mechanical energy E=K+U of the system, where U is the sum of the gravitational potential energy and the elastic potential energy
Force and Potential Energy
Force and Potential Energy
We have studied in detail two specific conservative forces, gravitational force and elastic force.
We have seen there is a definite relationship between a conservative force and the corresponding potential energy function.
The force on a mass in a uniform gravitational field is Fy = - mg. The corresponding potential energy function is U(y) = mgy.
The force exerted on a body by a spring of force constant k is Fx = - kx. The corresponding potential energy function is Us(x) = (1/2)kx2.
In some situations, you are given an expression for potential energy as a function of position and have to find corresponding force.
Force and Potential Energy
Consider motion along a straight line, with coordinate x Fx(x) is the x-component of force as function of x U(x) is the potential energy as function of x Work done by conservative force equals the negative of the change
U in potential energy: UW
For infinitesimal displacement x, the work done by force Fx(x) during this displacement is ~ Fx(x)x (suppose that this interval is so small that the force will vary just a little)
In the limit x0:
UxxFx )(x
UxFx
)(
dx
xdUxFx
)()( Force from potential
energy, one dimension
Force and Potential Energy
In regions where U(x) changes most rapidly with x (i.e. where dU(x)/dx is large) the greatest amount of work is done during the displacement, and it corresponds to a large force magnitude
When Fx(x) is in positive x-direction, U(x) decreases with increasing x
Thus, Fx(x) and U(x) have opposite sign
Thus, the force is proportional to the negative slope of the potential energy function
The physical meaning: conservative force always acts to push the system toward lower potential energy
dx
xdUxFx
)()( Force from potential
energy, one dimension
Force and Potential Energy
Lets verify if this expression correctly gives the gravitational force and the elastic force when using the gravitational potential energy and the elastic potential energy:
2
2
1)( kxxU kxkx
dx
d
dx
xdUxFx
2
2
1)()(
mgyyU )( mgmgydy
d
dy
ydUxFy
)()(
The gravitational potential energy is linearly related to the elevation (i.e. constant slope) and the force is constant.
The elastic potential energy varies quadratically with position. The force varies in a linearly.
Force and Potential Energy
Force and Potential Energy in 3D
Conservative force in three dimensions has components Fx, Fy, and Fz Each component may be function of coordinates x, y, z
Potential energy change U is the function of coordinates as well When particle moves a small distance x in x-direction, the force Fx is
~constant. It does NOT depend on Fy and Fz because these components of force are perpendicular to the displacement and do NO work
x
UFx
y
UFy
z
UFz
0
0
0
z
y
x
dx
dUFx
dy
dUFy
dz
dUFz
Force from potential energy, three dimensions
k
dz
dUj
dy
dUi
dx
dUF ˆˆˆ
UF
Energy Diagrams
Energy Diagrams
In situations where a particle moves in one-dimension only under influence of a single conservative force it is very useful to study the graph of the potential energy as a function of position U(x)
At any point on a graph of U(x), the force can be calculated as the negative of the slope of the potential energy function
Fx = - dU/dx
Example: Glider on an air track Spring exerts a force Fx=-kx Potential energy function U(x) Limits of the motion are the points
where U curve intersects the horizontal line representing the total mechanical energy E
Energy Diagrams
Any point where the force is zero is called equilibrium point
These are the "critical points" on the graph of U(x):
Points on the graph that are local minima correspond to "stable equilibria" since the force on particle tends to push it back toward the equilibrium point.
Points on the graph that are local maxima correspond to "unstable equilibria" since force on particle tends to push it back toward the equilibrium point.
Points on the graph that are inflection points correspond to "neutral equilibria".
If the total mechanical energy is known, then the potential energy graph can be used to determine the speed at any point since E = K + U is constant (i.e. use K = E – U and then find speed)
Energy Diagrams
Bounds of the Motion
x
y
A Pendulum
R
-10
0
10
20
30
40
50
-4 -3 -2 -1 0 1 2 3 4
U(x
) (J
)
x (m)
)cos(1(
)sin(
Ry
Rx
mgyU
-10
0
10
20
30
40
50
-4 -3 -2 -1 0 1 2 3 4
E a
nd
U(x
) (J
)
x (m)
-10
0
10
20
30
40
50
-4 -3 -2 -1 0 1 2 3 4
E, K
(x)
and
U(x
) (J
)
x (m)
+ m g y m v =
E = K + U
2
2
1
What is the motion?
K can never be negative
Motion is boundedTurning Points