7. Conservation of Energy

1. Conservative & Non-conservative Forces

2. Potential Energy

3. Conservation of Mechanical Energy

4. Potential Energy Curves

How many different energy conversions take place as the Yellowstone River plunges over Yellowstone Falls?

P.E. K.E., sound, & heat

Work against conservative force (path independent),

e.g., gravity:

Work can be stored potential energy.

Work against non-conservative force (path

dependent),

e.g., friction:

Work is dissipated heat & entropy.

7.1. Conservative & Non-conservative ForcesF is conservative if

0C

d F r for any closed path C.

B

AB AW d F r is path-independent

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

Gravity is conservative.

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 = m g L 0

Friction is non-conservative.

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

Example 7.1. Riding the Elevator

A 55 kg engineer takes an elevator from her office on the 33rd floor to the 59th floor.

Later, she descends to street level.

If she takes her office as the zero of potential energy,

and if the distance between floors is 3.5 m,

what’s her potential energy

(a) in her office.

(b) on the 59th floor.

(c) at street level ?

(a) 33 0U

(b) 59 59 33U m g h h 255 9.8 / 3.5 59 33kg m s m 49 kJ

(c) 1 1 33U m g h h 255 9.8 / 3.5 1 33kg m s m 60 kJ

Application: Pumped Storage

Northfield Mountain Pumped Storage Project, MA, USA

Excess electric energy stored by pumping water to higher ground.(see Prob. 29)

Elastic Potential Energy

2

1

x

xU F x d x

F k xIdeal spring:

2

1

x

xk x d x 2 2

2 1

1

2k x x

0 0U at x

21

2U x k x parabolic

Example 7.2. Springs vs Gasoline

A car’s suspension consists of springs with overall effective k = 120 kN/m.

How much these springs need be compressed to store the same amount

of energy as in 1 gram of gasoline?

From Appendix C: Energy contents of gasoline = 44 MJ / kg.

3 3 2110 44 10 / 120 /

2kg kJ kg kN m x

2 211

15x m 0.86x m 86 cm

Springs can’t compete with gasoline as energy source.

Example 7.3. Climbing Rope

Climbing ropes are springy to cushion falls.

Consider rope with F = k x + b x2 , where k = 223 N/m, b = 4.10 N/m2.

Find the potential energy when it’s stretched 2.62 m, taking U = 0 at x = 0.

2

0

xU k x b x d x 2 3

0

1 1

2 3

x

k x b x

2 31 1

2 3k x b x

2 321 1223 / 2.62 4.10 / 2.62

2 3N m m N m m

741 J

which is only 3% of U = ½ k x2 .

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 7.4. Tranquilizing an Elephant

A biologist uses a spring-loaded gun to shoot tranquilizer darts into an elephant.

The gun’s spring has k = 940 N/m, & is compressed x = 25 cm before firing a 38-g dart.

Assuming the gun points horizontally, at what speed does the dart leave the gun?

Initial state: 2

0 0

1

2E U k x

Final state: 21

2E K m v

kv x

m 0E E

23

940 /25 10

38 10

N mv m

kg

39 /m s

Problem is harder to solve by 2nd law.

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

GOT IT? 7.3. Bowling Ball

A bowling ball is tied to the end of a long rope and suspended from the

ceiling.

A student holds the ball to her nose, then releases it from rest.

Should she duck as it swings back?

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

0 ncE E W

m g x

Final state: 0E

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 . ( 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

GOT IT?. 7.4.

Below is the potential energy curve for an electron in a microelectronic device.

(a)Find the point where the force on the electron is greatest.

(b) Find the rightmost position possible if the electron has total energy E1 .

(c) Find the leftmost position possible if the electron has total energy E2 & starts out to the right of D.

(d) Find a point where the force on the electron is zero.

(e) Find a point where the force on the electron poinjts to the left.

In some cases, there may be multiple answers.

(E)

(B)

(A,D)

(C)

(B,E)