Short Version : 7. Conservation of Energy

7.1. Conservative & Non-conservative Forces

F is conservative if

0C

d F r for every closed path C.

B

AB AW d F r is path-independenti.e.

,

0C

d F r

B

AB AW d F r is path-dependent

WAB WBA + WAB = 0 WAB = WBA = WAB

F is non-conservative if there is a closed path C such that

WBAWAB

Mathematica

Example: Work done on climber by gravity

Going up:

W1 = ( m g ) h

= m g h

Going down:

W2 = ( m g ) ( h)

= m g h

Round trip: W = W1 + W2 = 0

Horizontal displacement requires no work.

Gravity is conservative.

ˆm gF j

ˆh r j

ˆh r j

ˆm gF j

Example: Work done on trunk by friction

Going right:

W1 = ( m g ) L = m g L

Going left:

W2 = ( m g ) ( L) = m g L

Round trip: W = W1 + W2 = 2 m g L 0

Friction is non-conservative.

ˆL r iˆf m gF i

ˆL r i ˆf m gF i

GOT IT? 7.1.

If it takes the same amount of work to push a trunk across a rough floor as it

does to lift a weight to the same distance straight upward.

How do the amounts of work compare if the trunk & weight are moved along

curved paths between the same starting & end points?

Ans. Work is greater for the trunk.

7.2. Potential Energy

Conservative force:

Potential energy = stored work

= ( work done by force )

B

AB AB AU W d F r

Note: only difference of potential energy matters.

1-D case: 2

1

x

xU F x d x

Constant F: 2 1U F x x

Gravitational Potential Energy

U m g h

Horizontal component of path does not contribute.

Vertical lift: mgh

U mg y

m g

Elastic Potential Energy

2

1

x

xU F x d x

Ideal spring:

2

10

x

xk x x d x

2

1

20

1

2

x

x

k x x x

00U at x x

0

2

0 0

1

2

x

xU x k x x d x k x x

parabolic

x x0 x = x0 x x0

U is always positive

F k x 0k x x x0 = equilibrium position

2 22 1 0 2 1

1

2k x x x x x 2 2

2 0 1 0

1

2k x x x x

Let

Setting x0 = 0 : 21

2U x k x

21

2k x

7.3. Conservation of Mechanical Energy

netK W c ncW W ncU W

ncE K U W

Mechanical energy: E K U

Law of Conservation of Mechanical Energy:

0E K U if 0ncW ( no non-conservative forces )

constantE K U

Example 7.5. Spring & Gravity

A 50-g block is placed against a spring at the bottom of a frictionless slope.

The spring has k = 140 N/m and is compressed 11 cm.

When the block is released, how high up the slope does it rise?

Initial state: 20 0 0

1

2E U k x

Final state: E U m g h

20

2

k xh

m g

0E E

22

3 2

140 / 11 10

2 50 10 9.8 /

N m mh

kg m s

1.7 m

Example 7.6. Sliding Block

A block of mass m is launched from a spring of constant k that is compressed a distance x0.

The block then slides on a horizontal surface of frictional coefficient .

How far does the block slide before coming to rest?

Initial state: 20 0 0

1

2E U k x

Work done against friction:

nc fW f x

20

2

k xx

m g

m g x

Final state: 0E

Launch:

21 1

1

2E K m v

1 0 0E E Conservation of energy :

10 ncE W

7.4. Potential Energy Curves

Frictionless roller-coaster track

How fast must a car be coasting at

point A if it’s to reach point D?

A CE UCriterion:

21

2 A A Cm v m g h m g h

2A C Av g h h

turning points

potential barrier

potential well

Example 7.7. H2

Near the bottom of the potential well of H2, U = U0 + a ( x x0 )2 ,

where U0 = 0.760 aJ, a = 286 aJ / nm2 , x0 = 0.0741 nm. ( 1 aJ = 1018 J )

What range of atomic separation is allowed if the total energy is 0.717 aJ?

Turning points:

E U 2

0 0U a x x

00

E Ux x

a

2

0.717 0.760

286 /

aJ aJ

aJ nm

0.0123 nm

0.0864 0.0618x nm to nm0 0.074x nm

Force & Potential Energy

Force ~ slope of potential curve

U F r F x

( x along direction of F )

UF

x

0limx

UF

x

dU

d x

dU F r

Gaussian Gun

1 2

1 2U U U

1 1 2

K K U UK

i induced i iU M r H r

12U U

2 1U U

Assume fields of the induced dipoles negligible compared to that of the magnet.

3

1H

r

Video

1 2E K U U

12E K U

i iH H

2 1 2 1r r U U