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Ch7 Work and Energy


Ch.7 Work and Energy
In the weight-lifting competition of the 1996 Olympics, Andrey lifted a record-breaking 260kg from the floor to over his head (about 2m). In 1957 Paul stooped beneat

h a reinforced wood platform and then pushed upward on the platform with his back, lifting the platform and its load about 1cm.The composite weight of the load was 27,900N!

Who did more work on the objects he lifted-Andrey or Paul?

Ch.7 Work and Energy
7.1 Work Done by a Constant Force 7.2 Scalar Product of Two Vectors 7.3 Work Done by a Varying Force 7.4 Kinetic Energy and the Work-Energy Principle 7.5* Kinetic Energy at Very High Speed

7.1 Work and Energy (1’36”)

Work Done by a Constant Force
? W ? F ? r cos ?
W ? Fd cos ?

? ? ?r ? d

Example 1
(a) W ? Fd cos ? ? 0 (negative work)
(b) W ? Fd cos

? ? W ? F ?d
? F

?
2

?0

?

? v
? d

(zero work)

(c)

W ? Fd cos ? ? 0 (positive work)

(d)

W ? Fd cos 0 ? Fd (max. work)

7.2 Scalar Product of Two Vectors
The Scalar Product ( Dot Product)
? ? A ? B ? AB cos ?
? B

?

? A

? ? a ? b ? ab cos ?

? ?
?

?

? ? ? ? / 2,A ? B ? AB cos ? ? 0; ? ? ? ? / 2,A ? B ? AB cos ? ? 0; ? ? ? ? / 2,A ? B ? AB cos ? ? 0; ? ? ? ?,A ? B ? AB cos ? ? - AB;
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The Commutative Law
? ? A ? B ? AB cos ? ? BA cos ? ? ? ? ? A? B ? B ? A

? B

? ? ? ? A ? Ax i ? A y j ? Az k ? ? ? ? B ? Bx i ? B y j ? B z k ? ? A ? B ? Ax B x ? A y B y ? Az B z

?

? A

Example 2

? ? F1 ? d ? F1 d cos ? ? ? F1 d ? ? 1 0 F2 ? d ? F2 d cos 60 ? F2 d 2 ? ? ? F3 ? d ? F3 d cos ?0 2
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? ? F ? d ? Fd cos ?

7

The Scalar Product (Dot Product) 1’

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7.3 Work Done by a Varying Force
Work Done by a General Variable Force
? ? ? ? dW ? F ? d r ? F ? d s ? Fds cos ?
W ab ?

b
? dr

?

b a

dW ?
b

?

b a

? ? F ? dr ?

?

b

? F dr cos ?

a

Wab ?

?

? ? F ? ds ?

a

?

b

F cos ? ds

?
? r

?, r

? ?r
? F

a

If the force is constant
? F ? constant vector

a

? rb

then
W ab ?

?

b a

? ? F ? dr ? Fdr cos ? ? ? ? r?b ? ? ? ? ? ? F ? d r ? F ? ? ? d r ? F ? ( rb ? ra ) ? F ? ? r

? ra

o

?

ra

One-Dimensional Analysis In an xyz Set
? ? F ? F ( x )i

?W

j

? F j , avg ? x

d W ? F ( x ) dx
W ?

?

x xi

f

F ( x ) dx

Three-Dimensional Analysis In an xyz Set
W ab ? ? ?

?

b

? ? F ? dr ?

a

?

b

? ? F ? ds

a

? ? ? ? ? ds ? dr ? dxi ? dyj ? dzk
? ds

? ?

b

a

F x dx ? F y dy ? Fz dz F x dx ?

b

xb

xa

?

yb ya

F y dy ?

?

zb

?
? F

za

Fz dz

Example 3 (H. p.131)

a

? ? ? 2 Force F ? 3 x i ? 4 j ( N ), acts on a particle. How much work is done on the particle as it moves from coordinate s (2,3) to (3,0)?

Solution:
The work is

W ab ? ?

?
2

xb

xa

F x dx ?
2

? ?

yb ya

F y dy ?

?

zb

za

Fz dz

?

3

3 x dx ?

0

4dy ?

3

?

zb

0dz ? 7.0 ( J )

za

Work Done by a Spring Force
The Spring Force (Hook’s law)

? ? F ? ? kx

The Work Done by a Spring Force
Ws ? ?

?

b

? ? F ? dr

a

?

b

a

xf ? ? ? kx ? dx ? ? ? kxdx xi

? ?(

1 2

kx ?
2 f

1 2

kx )
1 2 kx i )
2

2 i

W s ? ?(

1 2

kx

2 f

?

Example 4 (H.10-p.137) A force acts on a 3.0kg particle-like object in such a way that the position of the object as a function of time is given by x = 3.0t-4.0t2+1.0t3 (SI). Find the work done on the object by the force from t=0 to t=4s. Solution: The force on the object is The work is
W ?
F ( t ) ? ma ? m d x dt
2 2

? ?8.0 ? 6.0t

?
xf

xf

F ( x )dx ?

xi

?

xf

F ( t )dx
2

xi

x = 3.0t-4.0t2+1.0t3
W ?

dx ? (3.0 ? 8.0t ? 3.0t )dt

?

F ( t )dx
4

xi

?

?

( ?8.0 ? 6.0t )(3.0 ? 8.0t ? 3.0t )dt
2

0

? 528( J ) ? 5.3 ? 10 ( J )
2

Work Done by Net Force
Wnet ?

?

b

a

? ? Fnet ? dr ?

?

b

a

? ? ? Fi ? dr ?
i

??

b

a

? ? Fi ? dr

Wnet ?

?W

Example 5 W1= -F1d W2= F2dcos600 = (1/2)F2d W3= 0 Wnet=W1+ W2 +W3 = -F1d + (1/2)F3d

7.4 Kinetic Energy & the Work-Energy Principle (30”) Energy

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Kinetic Energy K
Some Orders of Magnitude for K

The kinetic energy associated System K(J) with the state of motion of an Molecule of air 10-21(10-2eV) at room temperature object is
Electron in orbit ( 10Mev) around a nucleus Falling raindrop 10-18

10-3 103 105 1017

SI: J (Joule); 1erg = 10-7J 1eV = 1.6022?10-19J

A running human Automobile On a highway Large earthquake

Earth in orbital 1033 motion around the Sun

Kinetic Energy K

(9”)

Kinetic Energy K (40”)

Work-Kinetic Energy Principle
W ab ?

?

b

a

? ? Fnet ? dr

? ? ?

? ? ?

b

a b

Fnet

? cos ? dr ?

?

b

a

? F? dr
? r

? dr

?

?
?, r

? vb
b
? F

a b

ma ? ds ? mv dv ? 1 2
f

?

b

m
2 b

dv dt ?

?
? va

ds 1 2 mv a
2

a

mv

a

a
? d r ? ds ? vdt

o

W net ? K

? K i ? ?K

(Work-Kinetic Energy Principle)

The net work done on an object is equal to the change in its kinetic energy Work W is energy transferred to or from an object by means of a force acting on the object.

Work-Kinetic Energy Principle (1’23”)

Example 6 (H.p.124) An initially stationary 15.0kg crate of cheese wheels is pulled, via a cable, a distance d=5.70m up a frictionless ramp, to a height h of 2.50m, where it stops. (a) Wg = ? (b) WT = ?
? ? (a) W ? F ? d ? mgd cos ? g g ? ? mgd sin? ? ? mgh ? ?368J

Solution:

(b)The kinetic energy theorem gives us or thus

WT ? W g ? W N ? ?K

WT ? 368J ? 0 ? 0

WT ? 368J .

Example 7 (H.p.137-10P) A force acts on a 3.0kg object in such a way

that its position as function of time is given by x=3.0t-4.0t2 +1.0t3 . Find the work W by the force from t=0 to t=4s. Solution: By the work-kinetic energy theorem
W ? K
f

? Ki ?

1 2

mv

2 f

?

1 2

mv

2 i

Since the speed of the object is
v? dx dt ? 3.0 ? 8.0t ? 3.0t
2

At t=0s, vi=3.0m/s ;

and at t=4s, vf=19m/s
1 2

The work done is
W ? 1 2 mv
2 f

?

mv

2 i

? 5.3 ? 10 J
2

7.5* Kinetic Energy at Very High Speed
The Relativistic Kinetic Energy
For a particle,
0 ? v, ?velocity : ? ? kinetic energy : 0 - K
2

K ? m 0c (

1 1? v /c
2 2

? 1)

(relativistic kinetic energy)

?? ? ?

v ??c

K ?

1 2

m 0v

2

(classical kinetic energy)

Newton’s 2nd Law in Relativity
? ? dp F ? dt

p ? ? m0v

mv ? ? m0 ?

m0 1? v c
2 2

The Relativistic Kinetic Energy For a particle,
velocity : 0 ? v, kinetic energy : 0 - K

Using the work- energy theorem
K ? WF ?

?K ? W F
v

? Fds ? ?
L

dp dt

ds ?

L

?

vdp

0

The Relativistic Kinetic Energy

The Relativistic Kinetic Energy
K ?

?

v

0

vdp ? ? vd ( m v v ) ? m v v
0 2

v

2

v 0

? ? m v vdv
0

v

? mv v ? mv v

??

v

m 0 vdv 1?v /c
2 2 2 2 v

0

2

? m0 c

1?v /c

2 0

? mv c ? m0 c
2

2

m

v

? ? m

0

?

m

0 2

1? ?

K ? m vc

2

? m0c

2

?? ? ?

v ??c

K ?

1 2

m 0v

2

(Relativistic kinetic energy)

(classical kinetic energy)

Summary
Work Done by a General Variable Force
? ? ? ? dW ? F ? d r ? F ? d s ? Fds cos ?
W ab ?

?

b a

dW ?

?

b a

? ? F ? dr

Three-Dimensional Analysis In an xyz Set
W ab ?

?

b

? ? F ? dr ?

a

?

b

? ? F ? ds ?

a

?

xb

xa

F x dx ?

?

yb ya

F y dy ?

?

zb

za

Fz dz

Kinetic Energy K Work-Kinetic Energy Principle W net ? K The Relativistic Kinetic Energy
K ? m 0c (
2

f

? K i ? ?K
1 1? v /c
2 2

? 1)

Questions & Problems (Ch.7)
P.164: 38. 64, 67


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