Physics 211: Lecture 9 and 10 combined Todays Agenda

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Physics 211 Lecture 9 and 10 combinedTodays
Agenda

• Work Energy
• Discussion
• Definition
• Dot Product
• Work of a constant force
• Work/kinetic energy theorem
• Work of multiple constant forces
• Work done by variable forces
• Spring
• 3-D generalization to all this
• Conservative forces and potential energy
• Comments

Ill work out the Spring 2006 HE1 tonight from
7-9pm in Lincoln Hall Theater You can download it
from the course web page
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test this Thursday eveningwhere are you supposed
to be?
bring your I-card

• Constant acceleration problems (including
  multiparticle)
• Centripetal acceleration
• Inertial reference frames and transformations
• Newtons Laws of motion
• Newtons Law of Gravity
• Friction static and dynamic
• Free body diagrams, itemizing forces
• Tension in ropes (a.k.a. lines)
• Many body systems
• Atwoods machine, etc
• Springs

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Today show that net work equals change in
kinetic energy

• What is work?
• What is kinetic energy?
• Strategy
• Use Newtons laws to derive result mathematically
• Interpret result (work, kinetic energy)
• All this comes from Newton math
• No new assumptions
• A lot of logic
• Work is tied to forces
• Kinetic energy is tied to object (its mass,
  velocity)

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Definition of Work
Ingredients Force (F), displacement (?r) Work,
W, of a constant force F acting through a
displacement ?r is W F? ?r F ?r cos ? Fr
?r
F
?r
?
Fr
displacement
Dot Product
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Definition of Work...
Hairdryer

• Only the component of F along the displacement is
  doing work.
• Example cart on track

F
? r
?
F cos ?
W F? ?r F ?r cos q Fr ?r
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Aside Dot Product (or Scalar Product)
Definition a.b ab cos ? ab cos ? aba
ba cos ? bab Some properties a?b
b?a q(a?b) (qb)?a b?(qa) (q is a
scalar) a?(b c) (a?b) (a?c) (c is a
vector) Units multiply The dot product of
perpendicular vectors is 0 !!
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Aside Examples of dot products
i . i j . j k . k 1 i . j j . k k . i
0
a . b (1 i 2 j 3 k ).(4 i - 5 j 6 k )
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Aside Properties of dot products

• Magnitude
• a2 a2 a . a
• (ax i ay j) . (ax i ay j)
• ax 2(i . i) ay 2(j . j) 2ax ay (i . j)
• ax 2 ay 2
• Pythagorean Theorem!!

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Aside Properties of dot products

• Components
• a ax i ay j az k (ax , ay , az) (a . i,
  a . j, a . k)
• Derivatives
• Apply to velocity
• So if v is constant (like for UCM)

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Back to the definition of Work
Skateboard
Work, W, of a force F acting through a
displacement ? r is W F? ? r
F
? r
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Lecture 9, Act 1Work Energy

• A box is pulled up a rough (m gt 0) incline by a
  rope-pulley-weight arrangement as shown below.
• How many forces are doing work on the box?

(a) 2 (b) 3 (c) 4
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Lecture 9, Act 1Solution
N

• Draw FBD of box

T

• Any force not perpendicularto the motion will do
  work

f
N does no work (perp. to v)
T does positive work
mg
f does negative work
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Work 1-D Example (constant force)

• A force F 10 N pushes a box across a
  frictionlessfloor for a distance ?x 5 m.

F
?x
Work done by F on box WF F??x F ?x
(since F is parallel to ?x) WF (10 N) x
(5 m) 50 Joules (J)
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Units

• Force x Distance Work

Newton x ML / T2
Meter Joule L ML2 / T2
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Work Kinetic Energy

• A force F 10 N pushes a box across a
  frictionlessfloor for a distance ?x 5 m. The
  speed of the box is v1 before the push and v2
  after the push.

v1
v2
F
m
i
?x
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Work Kinetic Energy...

• Since the force F is constant, acceleration a
  will be constant. We have shown that for
  constant a
• v22 - v12 2a(x2-x1) 2a?x.
• multiply by 1/2m 1/2mv22 - 1/2mv12 ma?x
• But F ma 1/2mv22 - 1/2mv12 F?x

a
m
i
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Work Kinetic Energy...

• So we find that
• 1/2mv22 - 1/2mv12 F?x WF
• Define Kinetic Energy K K 1/2mv2
• K2 - K1 WF
• WF ?K (Work/kinetic energy theorem)

v2
v1
a
m
i
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Work/Kinetic Energy Theorem

• Net Work done on object
• change in kinetic energy of object

• Well prove this same thing for a variable force
  in a bit.

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Lecture 9, Act 2Work Energy

• Two blocks have masses m1 and m2, where m1 gt m2.
  They are sliding on a frictionless floor and have
  the same kinetic energy when they encounter a
  long rough stretch (i.e. m gt 0) which slows them
  down to a stop. They experience same m
• (Which one will go farther before stopping?

(a) m1 (b) m2 (c) they will go the same
distance
m1
m2
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Lecture 9, Act 2Solution

• The work-energy theorem says that for any object
  WNET DK
• In this example the only force that does work is
  friction (since both N and mg are perpendicular
  to the blocks motion).

N
f
m
mg
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Lecture 9, Act 2 Solution

• The work-energy theorem says that for any object
  WNET DK
• In this example the only force that does work is
  friction (since both N and mg are perpendicular
  to the blocks motion).
• The net work done to stop the box is - fD
  -mmgD.

• This work removes the kinetic energy that the
  box had
• WNET K2 - K1 0 - K1

m
D
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Lecture 9, Act 2 Solution

• The net work done to stop a box is - fD -mmgD.
• This work removes the kinetic energy that the
  box had
• WNET K2 - K1 0 - K1
• This is the same for both boxes (same starting
  kinetic energy).

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A simple applicationWork done by gravity on a
falling object

• What is the speed of an object after falling a
  distance H, assuming it starts at rest?
• Wg F? ?r mg ?r cos(0) mgH
• Wg mgH
• Work/Kinetic Energy Theorem
• Wg mgH 1/2mv2

v0 0
mg
j
?r
H
v
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Work done by gravity

• Wg F? ?r mg ?r cos ?
• -mg ?y
• (remember ?y yf - yi)
• Wg -mg ?y
• Depends only on ?y !
• not ?x

m
mg
?r
?
?y
m
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Work done by gravity taking a funny path...
W NET W1 W2 . . . Wn

• Depends only on ?y,
• not on path taken!
• This is a very artificial example
• Be careful of very artificial examples!

m
mg
?y
j
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What about multiple forces?
Suppose FNET F1 F2 and the displacement is
?r. The work done by each force is W1 F1? ?r
W2 F2 ? ?r WTOT W1 W2
F1? ?r F2? ?r (F1 F2 )? ?r
WTOT FTOT? ?r Its the total force
that matters!!
FNET
F1
?r
F2
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Comments

• Time interval not relevant
• Run up the stairs quickly or slowly...same W
• Since W F? ?r
• No work is done if
• F 0 or
• ?r 0 or
• ? 90o

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Comments...

• W F? ?r
• No work done if ? 90o.
• No work done by T.
• No work done by N.

T
v
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Lecture 10, Act 1Falling Objects
Falling objects

• Three objects of mass m begin at height h with
  velocity 0. One falls straight down, one slides
  down a frictionless inclined plane, and one
  swings on the end of a pendulum. What is the
  relationship between their velocities when they
  have fallen to height 0?

(a) Vf gt Vi gt Vp (b) Vf gt Vp gt Vi
(c) Vf Vp Vi
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Lecture 10, Act 1Solution
Only gravity will do work Wg mgH 1/2 mv22 -
1/2 mv12 1/2 mv22
does not depend on path !!
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Work done by Variable Force (1D)

• When the force was constant, we wrote W F ?x
• area under F vs. x plot
• For variable force, we find the areaby
  integrating
• dW F(x) dx.
• Remember integration is just
• sophisticated summing

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Work/Kinetic Energy Theorem for a Variable Force
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1-D Variable Force Example Spring

• For a spring we know that Fx -kx.

F(x)
x2
x1
x
relaxed position
-kx
F - k x1
F - k x2
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Spring...

• The work done by the spring Ws during a
  displacement from x1 to x2 is the area under the
  F(x) vs x plot between x1 and x2.

F(x)
x2
x1
x
Ws
relaxed position
-kx
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Spring...
Spring

• The work done by the spring Ws during a
  displacement from x1 to x2 is the area under the
  F(x) vs x plot between x1 and x2.

F(x)
x2
x1
x
Ws
-kx
In this example it is a negative number. The
spring does negative work on the mass
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Lecture 10, Act 2Work Energy

• 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 and
  its mass were halved, how far x2 would the spring
  compress ?

(a) (b)
(c)
x
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Lecture 10, Act 2Solution

• Again, use the fact that WNET DK.

In this case, WNET WSPRING -1/2 kx2 and
?K -1/2 mv2
so kx2 mv2
x1
v1
m1
m1
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Lecture 10, Act 2Solution
So if v2 2v1 and m2 m1/2
x2
v2
m2
m2
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Problem Spring pulls on mass.

• A spring (constant k) is stretched a distance d,
  and a mass m is hooked to its end. The mass is
  released (from rest). What is the speed of the
  mass when it returns to the relaxed position if
  it slides without friction?

m
relaxed position
stretched position (at rest)
m
d
after release
m
v
back at relaxed position
m
vr
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Problem Spring pulls on mass.

• First find the net work done on the mass during
  the motion from x d to x 0 (only due to the
  spring)

stretched position (at rest)
m
d
relaxed position
m
i
vr
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Problem Spring pulls on mass.

• Now find the change in kinetic energy of the mass

stretched position (at rest)
m
d
relaxed position
m
i
vr
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Problem Spring pulls on mass.

• Now use work kinetic-energy theorem Wnet WS
  ?K.

stretched position (at rest)
m
d
relaxed position
m
i
vr
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Work by variable force in 3-D

• Work dWF of a force F acting
• through an infinitesimal
• displacement ?r is
• dW F.?r
• The work of a big displacement through a variable
  force will be the integral of a set of
  infinitesimal displacements
• WTOT F.?r

ò
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Work/Kinetic Energy Theorem for a Variable Force
in 3D
Sum up F.dr along path Thats the work
integral That equals change in KE For
conservative forces, the work is path independent
and depends only on starting point and end point
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Work by variable force in 3-DNewtons
Gravitational Force

• Work dWg done on an object by gravity in a
  displacement dr isgiven by
• dWg Fg.dr (-GMm / R2 r).(dR r
  Rd???) dWg (-GMm / R2) dR (since
  r.? 0, r.r 1)

dR
??

r
dr
Rd?
Fg
d?
R
M
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Work by variable force in 3-DNewtons
Gravitational Force

• Integrate dWg to find the total work done by
  gravity in a bigdisplacement
• Wg dWg (-GMm / R2)
  dR GMm (1/R2 - 1/R1)

Fg(R2)
R2
Fg(R1)
R1
M
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Work by variable force in 3-DNewtons
Gravitational Force

• Work done depends only on R1 and R2, not on the
  path taken.

m
R2
R1
M
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Newtons Gravitational ForceNear the Earths
Surface

• Suppose R1 RE and R2 RE ?ybut we
  have learned thatSo Wg -mg?y

m
RE ?y
RE
M
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Conservative Forces

• In general, if the work done does not depend on
  the path taken (only depends the initial and
  final distances between objects), the force
  involved is said to be conservative.
• Gravity is a conservative force
• Gravity near the Earths surface
• A spring produces a conservative force

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Conservative Forces

• We have seen that the work done by a conservative
  force does not depend on the path taken.

W2
W1 W2
W1

• Therefore the work done in a closed path is 0.

W2
WNET W1 - W2 W1 - W1 0
W1
Potential energy change from one point to another
does not depend on path
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Recap of todays lecture

• Work Energy (Text 6-1 and 7-4)
• Dot Product (Text 6-2)
• Work of a constant force (Text 7-1 and 7-2)
• Work/kinetic energy theorem (Text 6-1)
• Work done by variable force (Text 6-1)
• Spring
• Work done by variable force in 3-D (Text 6-1)
• Newtons gravitational force (Text 11-2)
• Conservative Forces (Text 6-4)
• Look at textbook problems Chapter 6 1, 20,
  23, 24, 25, 26, 34 45