9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 1

PHY 113 A General Physics I 9-9:50 AM MWF Olin 101

Plan for Lecture 13:

Chapter 8 -- Conservation of energy

1. Potential and kinetic energy for conservative forces

2. Energy and non-conservative forces

3. Power

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 2

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 3

Note: Because of PHY 114 exams on Sunday and Monday, the tutorials those nights will be moved from Olin 101 – check for signs – this week only.

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 4

( ) ( )( )

( ) ( ) EUmvUmv

UUdW

mvmvdW

ffii

iftotalfi

iftotalfi

f

i

f

i

=+=+⇒

−−=⋅=

−=⋅=

∫

∫

→

→

rr

rrrF

rF

r

r

r

r

22

22

21

21

:forces veconservatiFor

21

21

:oremenergy the kinetic- workGeneral

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 5

Example A block, initially at rest at a height h, slides down a frictionless incline. What is its final velocity?

h h=0.5m

i

f

( )

( ) 2

21 ;0 ;0 :

; ;0 :

fff

ii

mvEUvf

mghEmghUvi

==>

===

r

r

( )( )m/s 3.13

5.08.92

221 2

=

=

=

=

ghv

mghmv

f

f

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 6

Energy diagram

E

yi yf

K

U

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 7

Energy diagrams 22

21

21 kxmvE +=

2max2

1221

2max2

1

max

,0(0) :0 when :Note ,0

: when :Note

kxmvUx

kxEvxx

==

=

==

=

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 8

Example: Model potential energy function U(x) representing the attraction of two atoms

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 9

Example: Simplified version:

(Assume that the sandbag is massive enough to provide the tension in the rope and R=3m.) If the actor starts at vi=0 and θ=60o, what is vf?

( )( )( )m/s 5.4

5.0138.92221

21

)cos1(21

22

2

=

−==

=+=

−=+=

if

fff

ii

gyv

mvmgymvE

mgRmgymvE θ0

0

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 10

Example: Mass sliding on frictionless looping track

i

iclicker exercise: In order for the ball completes the loop at A, what must the value of h?

A. h=R B. h=2R C. h>2R D. Not enough

information.

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 11

i Example: Mass sliding on frictionless looping track

( )RmgmvE

mghmvE

A

i

22121

2

2

+=

+=

mgRmv

Rvmmgn

A

A

21

21

:Aat on track stayingfor Condition

2

2

=⇒

−=−−0

0

( )

Rh

mgRmgRmgRRmgmvmghE A

25

252

212

21 2

=⇒

=+=+==

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 12

Another example; first without friction

Mass m1 (=0.2kg) slides horizontally on a frictionless table and is initially at rest. What is its velocity when it moves a distance ∆x=0.1m (and m2 (=0.3kg) falls ∆y=0.1m)?

iclicker exercise: What is the relationship of the final velocity of m1 and m2?

A. They are equal B. m2 is faster than m1. C. m1 is faster than m2.

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 13

Another example; first without friction

Mass m1 (=0.2kg) slides horizontally on a frictionless table and is initially at rest. What is its velocity when it moves a distance ∆x=0.1m (and m2 (=0.3kg) falls ∆y=0.1m)?

T

T

m2 g

21

2

21

21

221

21

mmxgmv

vmvmxTW

f

if

+∆

=

−=∆=0

xgmm

mmxTW

gmm

mmT

amgmTamT

∆+

=∆=

+=⇒

−=−=

21

21

21

21

22

1

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 14

Another example; now with friction

Mass m1 (=0.2kg) slides horizontally on a table with kinetic friction and is initially at rest. What is its velocity when it moves a distance ∆x=0.1m (and m2 (=0.3kg) falls ∆y=0.1m)?

T

T

m2 g

( ) 21

21 2

121

if vmvmxfTW −=∆−=

0

f

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 15

( ) xfmm

mxgmm

mmxfTW

fmm

mgmm

mmT

amgmTamfT

∆+

−∆+

=∆−=

++

+=⇒

−=−=−

21

1

21

21

21

2

21

21

22

1

( )

gmfmm

xfxgmv

vmxfTW

k

f

f

1

21

2

21

2221

µ=+

∆−∆=

=∆−=

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 16

iclicker exercise: Assume a mass m starts at rest at A and moves on the frictionless surface as shown. At what position is the speed the largest?

A. A B. B C. C D. none of these.

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 17

Power

Watt(s)

(Joules) : Units

: thatNote

: workof change of rate timeas Defined

≡=

⋅=⋅=

⋅=

≡

P

dtd

dtdW

ddWdt

dW P

vFrF

rF

9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 18

( )( ) Watts106/3102 /3 ,102For

:motorby exertedPower

44

4

×=×=

=×=

=⋅=≡

smNPsmvNT

Tvdt

dW P vF