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Physics 111: Lecture 9 Todays Agenda

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Physics 211: Lecture 9, Pg 2. Do you think the Illini will get ... Dyne-cm (erg) = 10-7 J. BTU = 1054 J. calorie = 4.184 J. foot-lb = 1.356 J. eV = 1.6x10-19 J ... – PowerPoint PPT presentation

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Title: Physics 111: Lecture 9 Todays Agenda


1
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

2
Do you think the Illini will get a bowl game this
year
  • Definitely
  • Probably
  • Dont know or care
  • Probably not
  • Definitely not

3
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

4
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).

5
Mass Energy (but not in Physics 211)
  • Particle Physics

E 1010 eV
(a)
(b)
E MC2
M
( poof ! )
(c)
6
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-brakes-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...

7
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
8
Definition of Work...
Hairdryer
  • Only the component of F along the displacement is
    doing work.
  • Example Train on a track.

F
? r
?
F cos ?
9
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 !!
10
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
11
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!!

12
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)

13
Back to the definition of Work
Skateboard
Work, W, of a force F acting through a
displacement ? r is W F? ? r
F
? r
14
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
15
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
16
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)
17
Units
  • Force x Distance Work

Newton x ML / T2
Meter Joule L ML2 / T2
18
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
19
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
20
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
21
Work/Kinetic Energy Theorem
  • Net Work done on object
  • change in kinetic energy of object
  • Well prove that this works for a variable force
    as well.

22
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
23
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
24
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
25
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).

26
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
27
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
28
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

29
Comments...
  • W F? ?r
  • No work done if ? 90o.
  • No work done by T.
  • No work done by N.

T
v
v
N
30
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
31
Lecture 9, Act 3Solution
  • First, draw all the forces in the system

32
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.

33
Recap of todays lecture
  • Work Energy
  • Discussion
  • Definition
  • Dot Product
  • Work of a constant force
  • Work/kinetic energy theorem
  • Properties (units, time independence, etc.)
  • Work of a multiple forces
  • Comments
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