Physics 111: Lecture 9 Todays Agenda

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Physics 111 Lecture 9Todays Agenda

• Work Energy
• Discussion
• Definition
• Dot Product
• Work of a constant force
• Work/kinetic energy theorem
• Work of multiple constant forces
• Comments

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Work Energy

• One of the most important concepts in physics
• Alternative approach to mechanics
• Many applications beyond mechanics
• Thermodynamics (movement of heat)
• Quantum mechanics...
• Very useful tools
• You will learn new (sometimes much easier) ways
  to solve problems

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Forms of Energy

• Kinetic Energy of motion.
• A car on the highway has kinetic energy.
• We have to remove this energy to stop it.
• The breaks of a car get HOT!
• This is an example of turning one form of energy
  into another (thermal energy).

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Mass Energy (but not in Physics 111)

• Particle Physics

E 1010 eV
(a)
(b)
E MC2
M
( poof ! )
(c)
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Energy Conservation
Wilberforce
Returning Can

• Energy cannot be destroyed or created.
• Just changed from one form to another.
• We say energy is conserved!
• True for any isolated system.
• i.e. when we put on the brakes, the kinetic
  energy of the car is turned into heat using
  friction in the brakes. The total energy of the
  car-breaks-road-atmosphere system is the same.
• The energy of the car alone is not conserved...
• It is reduced by the braking.
• Doing work on an isolated system will change
  its energy...

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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 Train on a track.

F
? r
?
F cos ?
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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) 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
Suppose
Then
a 1 i 2 j 3 k b 4 i - 5 j 6 k
a . b 1x4 2x(-5) 3x6 12 a . a 1x1
2x2 3x3 14 b . b 4x4 (-5)x(-5)
6x6 77
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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 for a variable force later.

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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.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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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
v
N
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Lecture 9, Act 3Work Energy

• An inclined plane is accelerating with constant
  acceleration a. A box resting on the plane is
  held in place by static friction. How many forces
  are doing work on the block?

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

• First, draw all the forces in the system

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Lecture 9, Act 3 Solution

• Recall that W F? ?r so only forces that have a
  component along the direction of the displacement
  are doing work.

FS
a
N
mg

• The answer is (b) 2.

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Recap of todays lecture

• Work Energy (Text 6-1 and 7-4)
• Discussion
• Definition (Text 6-1)
• Dot Product (Text 6-2)
• Work of a constant force (Text 7-1 and 7-2)
• Work/kinetic energy theorem (Text 6-1)
• Properties (units, time independence, etc.)
• Work of a multiple forces
• Comments
• Look at textbook problems Chapter 6 1, 39, 67