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AP Physics Chapter 8

Potential Energy

and Conservation

of Energy

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AP Physics

Turn in Chapter 7 Homework, Worksheet, and Lab

Lecture: Potential Energy and Conservation of Energy

Q&A

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Review on Work Done by Gravity

Work done by gravity (weight):

Object falling down from rest h

Interpretation:

cosW mgd mg h

Wg =

If you want to stop the object, K:

Object at height has the potential ability to do work (later)

< 0 Wg> 0

> 0 K > 0 (Isn’t obvious?)

+ 0 < 0

W < 0 by you You are doing _______ work

Object doing work to you

K

negative

K K

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Kinetic and Potential Energy

Kinetic energy: ability to do work due to motion– One object or system– Associated with an object

Potential energy: ability to do work due to a relative position– System

Object and earth Mass and spring

– Associated with a force

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Change in Potential Energy (U) and Work

The change of a potential energy is defined to equal to the negative of the work done by its associated force.

Not directly defining U W by gravity is the same for same h even

when object falls from different initial heights

U W

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Conservative and Nonconservative Forces

Conservative Force: The net work done by a conservative force on a particle moving

around any closed path is zero. The work done by a conservative force on a particle moving

between two points does not depend on the path taken by the particle.

Examples: gravity (weight), spring force

Nonconservative Force: The work done by a nonconservative force on a particle moving

between two points depends on the path taken by the particle. Example: friction, force you push a table from some initial point

to final point

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Force and Potential Energy

Only conservative forces have potential energies.

Nonconservative forces do not have potential energies associated with them.

Gravitational potential energy Spring potential energy

No pushing or pulling potential energy No frictional potential energy

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Gravitational Potential Energy, U or Ug

Define Gravitational Potential Energy:

Unit:

m: mass g = 9.8 m/s2

y: vertical position, height (upward has been defined as the positive direction.)

Reference point: y = 0 U = 0 Free to choose reference point. (U has no physical

significance, only U has physical significance.)

U

U mgy

U

m g y 2

mkg m

s

2m

kgs

N m J K

gW mg h mg h f img h h f imgh mgh

y

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Elastic or Spring Potential Energy, U or Usp

k: spring constant x: displacement from equilibrium (relaxed)

position of spring Reference point has been chosen to be at x = 0.

U

Define Spring Potential Energy: 21

2spU kx

Unit: spU

spW 2 21 1

2 2i fkx kx

2 21 1

2 2f ikx kx

2k x 2N

mm N m J W

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Example: Pg188-3

In Fig. 8-30, a 2.00 g ice flake is released from the edge of a hemispherical bowl whose radius r is 22.0 cm. The flake-bowl contact is frictionless.

a) How much work is done on the flake by its weight during the flake’s descent to the bottom of the bowl?

b) What is the change in the potential energy of the flake-Earth system during that descent?

c) If that potential energy is taken to be zero at the bottom of the bowl, what is its value when the flake is released?

d) If, instead, the potential energy is taken to be zero at the release point, what is its value when the flake reaches the bottom of the bowl?

r

Ice flake

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Solution: Pg188-3

mrkgm 220.0,1000.2 3

)a h

) ?b U

) 0.220 , ?c y r m U

)d y

W

U

U

U

0.220 , ?r m W

3 2 32.00 10 9.8 / 0.220 4.31 10mg h kg m s m J

34.31 10W J

3 2 32.00 10 9.8 / 0.220 4.31 10mgy kg m s m J

0.220 , ?r m U

3 2 32.00 10 9.8 / 0.220 4.31 10mgy kg m s m J

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Practice: Pg189-5

In Fig. 8-32, a frictionless roller coaster of mass m = 825 kg tops the first hill with speed v0 = 17.0 m/s at height h = 42.0 m. How much work does the gravitational force do on the car from that point to a) point A, b) point B, and c) point C? If the gravitational potential energy of the coaster-Earth system is taken to be zero at point C, what is its value when the coaster is at d) point B and e) point A? f) If mass m were doubled, would the change in the gravitational potential energy of the system between points A and B increase, decrease, or remain the same?

h

A

C

B

hh/2

v0O

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Solution: Pg189-5

) ,a O A h

) ,b O B h

) ,c O C h

825 , 17.0 / , 42.0om kg v m s h m

W

W

52

825 9.8 42.0 3.40 10m

W mg h mg h mgh kg m Js

0mg h

25

825 9.8 42.01.70 10

2 2 2

mkg mh mgh smg h mg J

0, ?W

, ?2 2

h hh W

0 , ?h h W

h

A

C

Bh

h/2

v0O

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Solution: Pg189-5 (continued)

) : , ?e A y h U 53.40 10U mgy mgh J

) : , ?2

hd B y U

U

) 2 , ?A Bf m m U

A BU

2 A Bm m U

51.70 102

hmgy mg J

B AU U 2 2

mgh mghmgh

2 (Doubled)A BU

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Practice: Pg189-6

A 1.50 kg snowball is fired from a cliff 12.5 m high with an initial velocity of 14.0 m/s, directed 41.0o above the horizontal.

a) How much work is done on the snowball by he gravitational force during its flight to the flat ground below the cliff?

b) What is the change in the gravitational potential energy of the snowball-Earth system during the flight?

c) If that gravitational potential energy is take to be zero at the height of the cliff, what is the value when the snowball reaches the ground?

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Solution Pg189-6 :

1.50 , 12.5 , 14.0 , 41.0oo o

mm kg h m v

s

)a h

) ?b U

)c y

W

U

U

21.50 9.8 12.5 184

mmg h kg m J

s

184W J

21.50 9.8 12.5 184

mmgy kg m J

s

12.5 , ?h m W

12.5 , ?h m U

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(Total) Mechanical Energy, E

Sum of kinetic and potential energy:

mecE K U

E K U

or simply,

g sK U U

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Conservation of Mechanical Energy

When only conservative forces are doing work within a system, the kinetic energy and potential energy can change. However, their sum, the mechanical energy E of the system, remains unchanged.

i fE E

i i f fK U K U 0E K U

K U

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y = 0

y = 1.5m + 11m

=12.5 m

y = 1.50 m

i

f

Let y = 0 after the net is stretched 1.5 m, then

fi EE 0 00 0

fsfgfisigi UUKUUK ....

fsig UU ..

. .s f g iU U 270.0 9.8 12.5 8575 8.58i

mmgy kg m J kJ

s

Example: Pg199-116A 70.0 kg man jumping from a widow lands in an elevated fire rescue net 11.0 m below the window. He momentarily stops when he has stretched the net by 1.50 m. Assuming that mechanical energy is conserved during this process and that the net functions like an ideal spring, find the elastic potential energy of the net when it is stretched by 1.50 m.

yi = 12.5 m, yf = 0, m = 70.0kg,

Us.f = ?

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i

f

y = 0

Let y = 0 at bottom, then

i fE E

0 0

2 21 1

2 2i i f fmv mgy mv mgy

21

2 i fmv mgy

iv

2 2i fv gy

22 2 9.8 160 56.f

m mgy m

s s

Practice: Pg197-96A volcanic ash flow is moving across horizontal ground when it encounters a 10o upslope. The front of the flow then travels 920 m on the upslope before stopping. Assume that the gases entrapped in the flow lift the flow and thus make the frictional force from the ground negligible; assume also that mechanical energy of the front of the flow is conserved. What was the initial speed of the front of the flow?

yi = 0, yf = 920m sin10o = 160m, vf = 0,

vi = ?

i iK U f fK U

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Practice: Pg191-31

A block with mass m = 2.00 kg is placed against a spring on a frictionless incline with angle = 30.0o (Fig. 8-43). (The block is not attached to the spring.) The spring, with spring constant 19.6 N/cm, is compress 20.0 cm and then released.

a) What is the elastic potential energy of the compressed spring?

b) What is the change in the gravitational potential energy of the block-Earth system as the block moves from the release point to its highest point on the incline?

c) How far along the incline is the highest point from the release point?

mk

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Solution: Pg191-31

2.00 , 19.6N

m kg kcm

100 cm 31.96 10 , 0.200

Nx m

m m

) sa U

) 0, 0, 0, ?i f f gb v v x U

) ?c L

y = 0hL

00

22 31 11.96 10 0.200 39.2

2 2

Nkx m J

m

s gK U U gU sU . .s f s iU U 39.2J

gW

sin

gU mg h mg h gU

2

39.22.00

2.00 9.8 /gU J

mmg kg m s

L

i •

f •

0 .s iU

h

h

L

2.00

4.00sin sin 30.0o

h mm

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y = 0

y = x

yi = x + hLet the maximum compression be x, and let y = 0 at maximum compression, then i

f iE E 00

2kx

22 2 4 2

2

mg mg k mghx

k

gf sfU U 21

2kx mg x h

2 2 2 0kx mgx mgh

22 2 8

2

mg mg kmgh

k

22mg mg kmgh

k

2

2 2 22.0 9.8 / 2.0 9.8 / 2 1960 / 2.0 9.8 / 0.40

1960 /

kg m s kg m s N m kg m s m

N m

0.10m

max

2.0 , 0.40 , 1960 ,

?

Nm kg h m k

mx

or 0.08m

Practice: Pg190-18A block of mass m = 2.0 kg is dropped from a height h = 40 cm onto a spring of spring constant k = 1960 N/m (Fig. 8-36). Find the maximum distance the spring is compressed.

h

k

gi siU U mgx mgh

2 2mgx mgh

f

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What if the block is slowly lowered to the top of the spring? How much will it be compressed?

Conservation of Energy?

No, hand is doing negative work when lowering the block.

Then how?

F 0

i

h

f

W

N = Fsp

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Force and Potential Energy

dUF

dx

For each potential energy, there is a conservative force associated with it.

Nonconservative forces have no potential energies.

F F x U W U

x

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Work Done By External Force

Work done by external force will change the total mechanical energy of the system:

ext mecW E

If there is also kinetic friction,

ext f mecW W E ext mec thW E E

where is the heat gain (change in thermal energy) f kthE W f d

Total Energy: inttot mec thE E E E

Eint = Chemical Energy & Nuclear Energy

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Work and Energy Change

ext mecW E

Total work done by __________________________.

W K

Total work done by _______________________________.

Work-Kinetic Energy Theorem:

all external forces (excluding spring force and gravity)

all external forces, normally including spring force and gravity.

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fW

. .3.0 , 15 , 20 , ?k ext th cube th floord m f F N E J E

.th floorE 15 3.0 20 25N m J J

. .th cube th floorE E thE

Practice: Pg198-105The temperature of a plastic cube is monitored while the cube is pushed 3.0 m across a floor at constant speed by a horizontal force of 15 N. The monitoring reveals that the thermal energy of the cube increases by 20 J. What is the increase in the thermal energy of the floor along which the cube slides?

.k th cubef d E

thE k kf d f d

. .th cube th floorE E kf d

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mecE

0.63 , 14 , 0, 8.1 , ?i f mec

mm kg v v y m E

s

21

2 img h mv

2

2

10.63 9.8 8.1 0.63 14 12

2

m mkg m kg J

s s

U K U

Practice: Pg198-111A 0.63 kg ball thrown directly upward with an initial speed of 14 m/s reaches a maximum height of 8.1 m. What is the change in the mechanical energy of the ball-Earth system during the ascent of the ball to that maximum height?

f iK K

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Practice: Pg193-51

In Fig. 8-51, a 3.5 kg block is accelerated from rest by a compressed spring of spring constant is 640 N/m. The block leaves the spring at the spring’s relaxed length and then travels over a horizontal floor with a coefficient of kinetic friction k = 0.25. The frictional forces stops the block in distance d = 7.8 m. What are

a) the increase in the thermal energy of the block-floor system,

b) the maximum kinetic energy of the block, and

c) the original compression distance of the spring.

No frictiond

(k)

1 2 3

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Solution: Pg193-51

) ?tha E

max) ?b K 1) ?c x

1 33.5 , 640 / , 0, 0.25, 7.8 , 0m kg k N m v d m v

thE 20.25 3.5 9.8 7.8 67.m

kg m Js

fW Nd mgd

During distance d (2 3),

W K 3 2K K 2K

2fW K

2K

max 2 67K K J

1 2U K

1x

21 2

1

2kx K

During uncompressing of spring, (1 2)

1 2E E 1 1U K 2 2U K

22 2 670.46

640 /

K Jm

k N m

No frictiond

(k)

1 2 3

fd

67fW J