Transcript Lecture4_Work_Proportions

on
IfWhat
this forces
trunk isact
moved
this storage
trunk?
across
the floor
at
constant velocity,
Push
Friction
Support of floor
(distributed among
the 4 casters it sits on)
Weight
1. the man’s PUSH must exceed the trunk’s
WEIGHT.
2. the man’s PUSH must exceed the total
friction forces.
3. all the above forces must exactly balance.
Forces all balance for constant velocity.
Imagine the work involved in sliding
this crate from the loading dock to
the center of the machine shop floor.
Compare that to the task of
pushing over to the far wall
(TWICE AS FAR)…
TWICE
AS MUCH
…or to the task of pushing WORK!
two identical crates together
to the center of the room
(TWICE AS MUCH).
Consider the work
involved in lifting
this heavy box to the
bottom storage shelf.
2h
compare this to
h
doing it twice (once for each box)
doing it once with BOTH boxes together
TWICE
lifting a single box to the
AS MUCH
upper shelf (TWICE AS FAR) WORK?
HALF
What about: half as high?
AS MUCH
half the weight? WORK?
Work  Force used in performing it.
Work  distance over which work is done.
Lifting weights is definitely work
even by this physics definition!
What about lowering weights back down?
When lowering an object, the force you apply to support
it (and keep it from dropping too fast) is NOT in the same
direction as its motion. In fact it is OPPOSITE.
Instead we think in terms of the weight of the object
(gravity’s pull downward) is moving you. When you
LIFT weights you do work on them. When you lower
them, they (gravity, actually) do work on you.
How much work
is being done
balancing this
tray in place?
What distance does
it move in the direction
of the force on it?
Work was done
picking this
briefcase up
from the floor.
How much work
is being done in
holding it still?
Does it move UP
at all in the direction
of the applied force?
How much work
does the cable do
in supporting
the bowling ball?
T
How much work does a crane do in
holding its load in
place above ground?
When holding this load steady in place,
how much energy must the crane’s motor
consume? Notice it could be shut off, and
hold the weight still. If no
energy is required there’s no real work
performed.
Work = Force  distance
If there is no
unbalanced
force, no
work is done!
If the pushed
object doesn’t even move
no work is done!
If the pushed object doesn’t move
in the direction of the force,
no work is being done on it!
If the object moves (despite an applied force) in a
direction opposite to the force, we say:
its doing work on the person trying to push
Or that
the person pushing does NEGATIVE work.
The crane lifts
its load up at
constant speed.
1.It lifts with a force > the load’s weight.
2.It lifts with a force = the load’s weight.
3.It lifts with a force < the load’s weight.
B
C
A
D
Work is done on the box during which stage(s)?
A. Lifting box up from floor.
B. Holding box above floor.
C. Carrying box forward across floor.
D. Setting box down gently to floor.
E. At all stages: A,B,C,D.
F. At stages A and C.
G. At stages A and D
H. At none of the stages illustrated.
Interlude:
Here’s a fairly
common trait
sought after
in the personals:
“Seeking…
“Weight proportional to height.”
What’s that supposed to mean?
Is that a desirable trait?
Which of the following geometric
shapes are “similar”?
A
B
1. A and C
3. D and F
5. C and G
C
D
E
F
G
2. A and C and G
4. A,C,E, and G
6. they are all polygons
and thus similar
A
B
C
D
E
Only C and D above are geometrically similar.
The geometric definition of “similar” requires more
than that the shapes are all triangles.
…more than that the triangles be the same “type”
isosceles triangles (A and E) or right triangles (B,C,D)
They need to “look alike”, but not be exactly alike
(that’s CONGRUENT, remember?)
1.35 inch
2.25 inch
C
D
1 inch
0.6 in
The dimensions of figure C are
in the same proportion as the
corresponding sides in figure D.
height of D 1.35

 0.60
height of C 2.25
width of D 0.6

 0.60
width of C 1.0
hypotenuse of D 1.477

 0.60
hypotenuse of C 2.462
anyway you look at it:
Triangle D is 0.6 the size of triangle C
C
1 inch
1.35 inch
2.25 inch
You may also remember
D
0.6 in
Ratios of any two sides within one figure
are in the same proportion
as the corresponding ratio of any similar figure.
height
1.35
 2.25
 2.25
width
0 .6
Each triangle is 2.25 as tall as it stands wide.
A description that applies equally to each.
hypotenuse
1
.
477
 1.094
 1.094
height
1.35
hypotenuse
1.477
 2.462
 2.462
width
0.60
Quantities are in proportion when they
simply scale with one another.
An object’s weight scales with its mass:
weight = 2.20462 lb/kg  mass
Similarly, the
weight of a liquid scales with its volume
weight = 2204.62 lb/m3  volume
density (of water)
The charge built up in a capacitor scales
with the voltage across its leads:
charge = C coul/volts  voltage
its capacitance
The cost of filling your tank scales with
the number of gallons of gasoline
cost = $2.34/gal  number of gallons
cost = $2.34/gal  number of gallons
Cost (in dollars)
$30
$25
$20
$15
$10
$5
5
10
15
Gallons (of gasoline)
Twice as much gas costs twice as much money!
Ten times as much gas costs 10 the money!
No gas costs nothing!
We say Cost  Gallons
Not all relationships are proportions:
oF
Temperature conversion follows:
9
F = C + 32
5
oC
y = mx + b
Notice: the graph of F vs C does not go through (0,0)!
This means while 0o C = 32o F
doubling (or tripling) both gives
0o C and 64o (or 96o) F which is
no kind of proportion!
also: not all relationships are even linear!
A little geometry:
A line cutting across a pair of parallel lines,
creates alternating congruent angles
A little geometry:
Whenever any two lines cross
the opposite angles formed are congruent.
Since the sum of all the “interior” angles
of every triangle add up to 360o…
A little geometry:
B
A
C
D F
E
…these two triangles are similar!
Look carefully to see which are the
“corresponding” (matching) sides.
Proportionalities
B
A
D F
E
C
The ratio of lengths of side A to side C
is in the same proportion as:
1.E/F
A
 ?
C
2.F/E
3.D/F
4.F/D
5.E/D
6.D/E
Proportionalities
A
B
D
C
A
 ?
z
1.A/w
2.A/x
3.B/x
4.C/x
5.D/x
6.D/w
y
w
x
z
s
a
r
The ratio of surface areas
A
 ?
a
1.R/r2
2. R2/r2
3. R2/r
4. R/r
5. r/R
R
A
S
Consider this block weighing “W”
height, h
weight, W
This stack
of 2 blocks height, 2h
weighs
how much?
Are these blocks
in proportion?
2W
To scale proportionally
height, h
weight, W
And this double-sized block weighs
1. 2W
2. 4W
3. 6W
4. 8W
5. 10W
6. 12W
More generally,
2L
2w
h
L
w
2h
original
=hwL
volume
new
=(2h)(2w)(2L)
volume
= ( 8 )hwL
=(8
original
)( volume )
Is weight meant to be proportional to height?
Weight  (Height)3
Each 1% increase in height should
correspond to a (1.01)3 = 1.03
3% increase in weight
5% increase in height
(5’4”  5’7”) 15.2% gain in weight
10% increase in height
(5’10”  6’5”) 30% gain in weight
SOME ANSWERS
Question 1
2. It lifts with a force We already demonstrated in class that I lifting requires,
= the load’s weight. on average, a force simply equal to a load’s weight.
Question 2
A only!
A. Lifting box up from floor. requires positive work
B. Holding box above floor. requires NO work
C. Carrying box forward across floor. No work if lifting force
perpendicular to direction of boxes motion!
D. Setting box down gently to floor. Involves NEGATIVE work!
Question 3
5. C and G
Question 4
4. F/D
Question 5
5. D/x
Question 6
2.
Question 7
4. 8W
“Similar” shapes have all their corresponding
(‘matching”)angles congruent, i.e., they can be lined up
so that all their corners are matched identically. It
makes all similar shapes “look alike” (like perfectly
scaled models of one another).
A, the shortest side of the larger triangle corresponds
to F, the shortest of the 2nd triangle. C is the middlesized side so corresponds to D.
A and z are bases of their respective (isosceles)
triangles, i.e, they are “corresponding sides.” D is
an “altitude” and so corresponds to x.
R2/r2
The ratio of the sides of the square areas S/s
= R/r since S is to R as s is to r.
However the areas A = S2 and a = s2.
Twice as wide…twice as tall…twice as thick…
means 222=8 times the volume (and mass).