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