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


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


rr

mGMF E

gravity ˆ2

ldFW gravitygravity



drrrdrrdrdrr

rdrdrrr

mGMldFdW E

gravity

0ˆˆ)ˆˆ(ˆ

)ˆˆ(ˆ2

rr

mGMF E

gravity ˆ2

ldFW gravitygravity



)11

(

1

1221

221

2

2

1

rrmGMW

drr

mGMdWW

drr

mGMdW

E

r

r

E

E

Clicker QuestionA. B. C.


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


12

11

rrmGMW e

Case 1 Case 2

Case 1:

2RE

RE

Same!

Clicker Question Solution


E

e

EEe R

mGM

RRmGMW

2

1

2

1

Case 2:E

e

Ee R

mGM

RmGMW

22

11

Potential Energy


Checkpoint Clicker Question A. B. C. D.


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



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


A)

B)

C)

RE

r

mGMrU e)(



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?



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?


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



A)

B)


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


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


)( hmgWgravity gdmWgravity 2

xx

x

N ldNW0

0

xx

x

N idxjmgW0

0

ˆˆ

00ˆˆ NWij

Work on Two Blocks


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


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


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


222

1fvmW

x

WT

xTxFW

2111, 2

1fnet vmKW

211 2

1fnet vmW xTFxFW netnet )(1or

Block Sliding


2

2

10

0

0

0

xkxdxkxdFWxx

x

xx

x

spring

m

Wv

mvW

springf

fspring

2

2

1 2

Block Sliding


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


mg

xkx

k

spring

friction

2

21

k

xmgx frictionk

spring

2

2

2

1xkWspring

Block Sliding 2


22

2

1

2

1fspring mvvmKW

22

2

1

2

1mvxkWspring

k

mvx f

2

Block Sliding 2


20

22

2

1

2

1vvmvmW fspring

mg

Wx

xmgW

k

friction

kfriction

Block Sliding 2


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


roughkfriction xmgW

Potential & Mechanical Energy


Work is Path Independent for Conservative Forces


Conservative Forces. Work is zero over closed path


Potential Energy


Gravitational Potential Energy


Mechanical Energy


Conservation of Mechanical Energy


NCspringsgravity WWW

Relax. There is nothing new here


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


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

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

Gravitational Potential Energy


Earth’s Escape Velocity


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

Potential Energy Function


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

Finding the potential energy change:


Use formulas to find the magnitude

Check the sign by understanding the problem…

Clicker Question A. B. C.


0% 0%0%

11 KhMghmgW

122 22 KKhMghmgW

Spring Potential Energy: Conserved


Vertical Springs


Massless spring

Vertical Spring in Gravitational Field


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


Non-conservative Forces


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

Summary


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


kx2

Spring Summary


x

M

Checkpoint


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


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


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.


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


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


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

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

Clicker Question A. B. C.


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


A=0 at x=0!

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

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

http://hyperphysics.phy-astr.gsu.edu/hbase/integ.html

Homework


Average=85% Average=86%

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

Example Test Problem


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


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

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