classical mechanics lecture 8 today’s examples a) work and kinetic energy problems today's...

Classical Mechanics Lecture 8

Today’s Examplesa) Work and Kinetic Energy Problems

Today's Concepts:a) Potential Energyb) Mechanical Energy

Mechanics Lecture 8, Slide 1

Midterm 2 will be held on March 13. Covers units 4-9

Main Points

Mechanics Lecture 7, Slide 2

Main Points

Mechanics Lecture 7, Slide 3

Main Points

Mechanics Lecture 7, Slide 4

Physics 211 Lecture 7, Slide 5

Work done by a Spring: Conservative Force!

Net Work over closed path = 0

Conservative Force

Work done by Gravitational Force

Mechanics Lecture 7, Slide 6

rr

mGMF E

gravity ˆ2

ldFW gravitygravity

Work done by Gravitational Force

Mechanics Lecture 7, Slide 7

drrrdrrdrdrr

rdrdrrr

mGMldFdW E

gravity

0ˆˆ)ˆˆ(ˆ

)ˆˆ(ˆ2

rr

mGMF E

gravity ˆ2

ldFW gravitygravity

Work done by Gravitational Force

Mechanics Lecture 7, Slide 8

)11

(

1

1221

221

2

2

1

rrmGMW

drr

mGMdWW

drr

mGMdW

E

r

r

E

E

Work done by Gravitational Force

Mechanics Lecture 7, Slide 9

Clicker QuestionA. B. C.

Mechanics Lecture 7, Slide 10

0% 0%0%

A) Case 1 B) Case 2 C) They are the same

In Case 1 we send an object from the surface of the earth to a height above the earth surface equal to one earth radius.

In Case 2 we start the same object a height of one earth radius above the surface of the earth and we send it infinitely far away.

In which case is the magnitude of the work done by the Earth’s gravity on the object biggest?

Case 1 Case 2

Mechanics Lecture 7, Slide 10

12

11

rrmGMW e

Case 1 Case 2

Case 1:

2RE

RE

Same!

Clicker Question Solution

Mechanics Lecture 7, Slide 11

E

e

EEe R

mGM

RRmGMW

2

1

2

1

Case 2:E

e

Ee R

mGM

RmGMW

22

11

Potential Energy

Mechanics Lecture 8, Slide 12

Checkpoint Clicker Question A. B. C. D.

Mechanics Lecture 8, Slide 13

0% 0%0%0%

In Case 1 we release an object from a height above the surface of the earth equal to 1 earth radius, and we measure its kinetic energy just before it hits the earth to be K1.

In Case 2 we release an object from a height above the surface of the earth equal to 2 earth radii, and we measure its kinetic energy just before it hits the earth to be K2.

Compare K1 and K2.

A) K2 = 2K1 B) K2 = 4K1 C) K2 = 4K1/3 D) K2 = 3K1/2

Case 1Case 2

wrong

Clicker QuestionA. B. C.

Mechanics Lecture 8, Slide 14

0% 0%0%

Usurface =

Usurface =

Usurface =

GMem0

GMem

RE

GMem

2RE

What is the potential energy of an object of mass m on the earths surface:

For gravity: +U0

Mechanics Lecture 8, Slide 14

A)

B)

C)

RE

r

mGMrU e)(

Clicker QuestionA. B. C.

Mechanics Lecture 8, Slide 15

0% 0%0%

A)

B)

C)

Case 1Case 2

RE

r

mGMrU e)(

E

e

R

mGMU 1

E

e

R

mGMU

21

E

e

R

mGMU

31

What is the potential energy of a object starting at the height of Case 1?

Clicker QuestionA. B. C.

Mechanics Lecture 8, Slide 16

0% 0%0%

What is the potential energy of a object starting at the height of Case 2?

A)

B)

C)

Case 1Case 2

RE

E

e

R

mGMU 2

E

e

R

mGMU

22

E

e

R

mGMU

32

r

mGMrU e)(

A)

B)

What is the change in potential in Case 1?

Mechanics Lecture 8, Slide 17

Case 1Case 2

RE

E

e

R

mGMU

21 E

e

R

mGMU

32 E

esurface R

mGMU

e

e

eeecase R

mGM

RRmGMU

2

11

2

11

e

e

eeecase R

mGM

RRmGMU

22

1

2

111

What is the change in potential in Case 2?

What is the change in potential in Case 2?

A)

B)

Mechanics Lecture 8, Slide 18

Case 1Case 2

RE

e

e

eeecase R

mGM

RRmGMU

3

21

3

12

E

e

R

mGMU

21 E

e

R

mGMU

32 E

esurface R

mGMU

e

e

eeecase R

mGM

RRmGMU

3

2

3

112

A) 2

B) 4

C) 4/3

D) 3/2

Case 1Case 2

Mechanics Lecture 8, Slide 19

What is the ratio

e

ecase R

mGMU

21 e

ecase R

mGMU

3

22

1

2

1

2

U

U

K

K

3

4

21

32

Work on Two Blocks

Mechanics Lecture 7, Slide 20

)( hmgWgravity gdmWgravity 2

xx

x

N ldNW0

0

xx

x

N idxjmgW0

0

ˆˆ

00ˆˆ NWij

Work on Two Blocks

Mechanics Lecture 7, Slide 21

KW

11, KWTension

x

WT

xTW

1

1

)( 2 hgmW 2221

220

2221 )(

2

1))((

2

1ff vmmvvmmK

22212 )(

2

1)( fvmmhgm

)(

2

21

221 mm

hgmvv ff

211

210

2111 )(

2

1)(

2

1ff vmvvmK

2111, 2

1ftension vmW

Work on Two Blocks

Mechanics Lecture 7, Slide 22

222

220

2222 )(

2

1)(

2

1ff vmvvmW

xx

x

Tension ldTW0

0

2,

jTT ˆ2

jdyld ˆ

02, TensionW

0)ˆˆ( TdyjjTdyldT

Work on Two Blocks 2

Mechanics Lecture 7, Slide 23

xFxdFWxx

x

0

0

00

0

xx

x

N xdNW

21

220

2

2

2

1

2

1

mm

Wv

mvvvmxFW

f

ff

Work on Two Blocks 2

Mechanics Lecture 7, Slide 24

222

1fvmW

x

WT

xTxFW

2111, 2

1fnet vmKW

211 2

1fnet vmW xTFxFW netnet )(1or

Block Sliding

Mechanics Lecture 7, Slide 25

2

2

10

0

0

0

xkxdxkxdFWxx

x

xx

x

spring

m

Wv

mvW

springf

fspring

2

2

1 2

Block Sliding

Mechanics Lecture 7, Slide 26

xmgxdimgxdFW k

xx

x

k

xx

x

frictionfriction

0

0

0

0

ˆ

)(2

1 20

2 vvmW ffriction

0

21

2 2

m

xkxmgv

springfrick

f

m

mvWv

friction

f

202

12

22

1springfrictionk xkxmg

mg

xkx

k

spring

friction

2

21

Block Sliding

Mechanics Lecture 7, Slide 27

mg

xkx

k

spring

friction

2

21

k

xmgx frictionk

spring

2

2

2

1xkWspring

Block Sliding 2

Mechanics Lecture 7, Slide 28

22

2

1

2

1fspring mvvmKW

22

2

1

2

1mvxkWspring

k

mvx f

2

Block Sliding 2

Mechanics Lecture 7, Slide 29

20

22

2

1

2

1vvmvmW fspring

mg

Wx

xmgW

k

friction

kfriction

Block Sliding 2

Mechanics Lecture 7, Slide 30

2patch

kfriction

xmgW

frictionspring WW

2)(2

1newspring xkW

2)(2

1

2 newpatch

k xkx

mg

k

xmgx patchk

new

2' patch

kpatchkfriction

xmgxmgW

2' kk

Block Sliding 2

Mechanics Lecture 7, Slide 31

roughkfriction xmgW

Potential & Mechanical Energy

Mechanics Lecture 8, Slide 32

Work is Path Independent for Conservative Forces

Mechanics Lecture 8, Slide 33

Work is Path Independent for Conservative Forces

Mechanics Lecture 8, Slide 34

Conservative Forces. Work is zero over closed path

Mechanics Lecture 8, Slide 35

Potential Energy

Mechanics Lecture 8, Slide 36

Gravitational Potential Energy

Mechanics Lecture 8, Slide 37

Mechanical Energy

Mechanics Lecture 8, Slide 38

Mechanical Energy

Mechanics Lecture 8, Slide 39

Conservation of Mechanical Energy

Mechanics Lecture 8, Slide 40

NCspringsgravity WWW

Relax. There is nothing new here

Mechanics Lecture 8, Slide 41

It’s just re-writing the work-KE theorem: everythingexcept gravityand springs

If other forces aren't doing work0E

NCWE NCWUK

NCspringsgravity WUUK

springsgravity UU totWK

Conservation of Mechanical Energy

Mechanics Lecture 8, Slide 42

http://www.statkraft.com/energy-sources/hydropower/pumped-storage-hydropower/

Pumped Storage Hydropower: Store energy by pumping water into upper reservoir at times when demand for electric power is low….Release water from upper reservoir to power turbines when needed ...

Energy “battery” using conservation of mechanical energy

Upper reservoir

Dh

Gravitational Potential Energy

Mechanics Lecture 8, Slide 43

Earth’s Escape Velocity

Mechanics Lecture 8, Slide 44

What is the escape velocity at the Schwarzchild radius of a black hole?

Potential Energy Function

Mechanics Lecture 8, Slide 45

U0 : can adjust potential by an arbitrary constant…that must be maintained for all calculations for the situation.

Finding the potential energy change:

Mechanics Lecture 8, Slide 46

Use formulas to find the magnitude

Check the sign by understanding the problem…

Clicker Question A. B. C.

Mechanics Lecture 8, Slide 47

0% 0%0%

11 KhMghmgW

122 22 KKhMghmgW

Spring Potential Energy: Conserved

Mechanics Lecture 8, Slide 48

Vertical Springs

Mechanics Lecture 8, Slide 49

Massless spring

Vertical Spring in Gravitational Field

Mechanics Lecture 8, Slide 50

Formula for potential energy of vertical spring in gravitational field has same form as long as displacement is measured w.r.t new equilibrium position!!!

Vertical Spring in Gravitational Field

Mechanics Lecture 8, Slide 51

Non-conservative Forces

Mechanics Lecture 8, Slide 52

Work performed by non-conservative forces depend on exact path.

Summary

Mechanics Lecture 8, Slide 53

Work done by any force other than gravity and springs will change E

Lecture 7Work – Kinetic Energy theoremtotalWK

Lecture 8For springs & gravity (conservative forces)

WU

Total Mechanical Energy E = Kinetic + Potential

UKE

NCWE

Summary

Mechanics Lecture 8, Slide 54

kx2

Spring Summary

Mechanics Lecture 8, Slide 55

x

M

Checkpoint

Mechanics Lecture 8, Slide 56

A. B. C. D.

Three balls of equal mass are fired simultaneously with equal speeds from the same height h above the ground. Ball 1 is fired straight up, ball 2 is fired straight down, and ball 3 is fired horizontally. Rank in order from largest to smallest their speeds v1, v2, and v3 just before each ball hits the ground.

A) v1 > v2 > v3

B) v3 > v2 > v1

C) v2 > v3 > v1

D) v1 = v2 = v3

1

2

3

h

87% correct

CheckPoint

Mechanics Lecture 8, Slide 57

A) v1 > v2 > v3

B) v3 > v2 > v1

C) v2 > v3 > v1

D) v1 = v2 = v3

1

2

3

h

They begin with the same height, so they have the same potential energy and change therein, resulting in the same change in kinetic energy and therefore the same speed when they hit the ground.

The total mechanical energy of all 3 masses are equal. When they reach the same height their kinetic energies must be equal, hence equal velocities.

0 UKE87% correct

Clicker Question

Mechanics Lecture 8, Slide 58

A. B. C.

0% 0%0%

Which of the following quantities are NOT the same for the three balls as they move from height h to the floor:

1

2

3

h

A) The change in their kinetic energiesB) The change in their potential energiesC) The time taken to hit the ground

Clicker CheckpointA. B. C.

Mechanics Lecture 8, Slide 59

0% 0%0%

A) B) C)

x

A box sliding on a horizontal frictionless surface runs into a fixed spring, compressing it a distance x1 from its relaxed position while momentarily coming to rest.

If the initial speed of the box were doubled, how far x2 would the spring compress?

12 2xx 12 2xx 12 4xx 90% correct

A) B) C)

21

2KE mv

21

2PE kx

CheckPoint

Mechanics Lecture 8, Slide 60

x

12 2xx 12 2xx 12 4xx

21

)2(

)(

)2(

)2()2(

)(

2

21

21

1

2

21

12

21

11

kvm

kvm

x

x

k

vmvvxx

k

mvvvxx

Mechanics Lecture 8, Slide 61

http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html#c4

Clicker Question A. B. C.

Mechanics Lecture 8, Slide 62

0% 0%0%

A) At xB) At oC) At y

A block attached to a spring is oscillating between point x (fully compressed) and point y (fully stretched). The spring is un-stretched at point o. At which point is the acceleration of the block zero?

x o y

Mechanics Lecture 8, Slide 63

A=0 at x=0!

http://hep.physics.indiana.edu/~rickv/SHO.html

Integrals as Area Under a curve

Mechanics Lecture 8, Slide 64http://hyperphysics.phy-astr.gsu.edu/hbase/integ.html

Homework

Mechanics Lecture 8, Slide 65

Average=85% Average=86%

Average=82% including 2 who did not do the homework

Example Test Problem

Mechanics Lecture 6, Slide 66

1. What is the work done by gravity during its slide to the bottom of the ramp?

2. What is the work done by friction during one pass through the rough spot?

3. What is the maximum distance the spring is compressed by the box?

4. What is the maximum height to which the box returns on the ramp?

Lecture Thoughts

Mechanics Lecture 8, Slide 67