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.

Weight

Support of floor (distributed amongthe 4 casters it sits on)

Friction

Push

What forces act on this storage trunk?If this trunk is moved across the floor at constant velocity,

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 ofpushing over to the far wall (TWICE AS FAR)…

…or to the task of pushing two identical crates together to the center of the room (TWICE AS MUCH).

TWICEAS MUCH

WORK!

Consider the work involved in liftingthis heavy box to the bottom storage shelf.

h

2h

doing it twice (once for each box)

lifting a single box to theupper shelf (TWICE AS FAR)

TWICEAS MUCH

WORK?

compare this to

doing it once with BOTH boxes together

What about: half as high? half the weight?

HALFAS MUCH

WORK?

Work Force used in performing it.

Work distance over which work is done.

Lifting weights is definitely workeven 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 youLIFT weights you do work on them. When you lower

them, they (gravity, actually) do work on you.

How much work is being donebalancing this tray in place?

Work was done picking this briefcase up from the floor. How much work is being done in holding it still?

What distance doesit move in the direction

of the force on it?

Does it move UPat all in the directionof 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 noenergy is required there’s no real work

performed.

Work = Force distance

If there is no unbalancedforce, no

work is done!

If the pushed object doesn’t even move

no work is done!

If the pushed object doesn’t movein the direction of the force,no work is being done on it!

If the object moves (despite an applied force) in adirection opposite to the force, we say:

its doing work on the person trying to pushOr that

the person pushing does NEGATIVE work.

The crane liftsits load up atconstant 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.

A

BC

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 DH. At none of the stages illustrated.

“Seeking…“Weight proportional to height.”

What’s that supposed to mean?

Here’s a fairly common trait sought after

in the personals:

Is that a desirable trait?

Interlude:

Which of the following geometricshapes are “similar”?

A B C D E F G

1. A and C 2. A and C and G3. D and F 4. A,C,E, and G5. C 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?)

The dimensions of figure C are in the same proportion as the

corresponding sides in figure D.

1 inch2.

25 in

ch

1.35

inch

0.6 in

2.462 inch

1.477 inch

60.025.2

35.1 C of height

D of height

60.00.1

6.0 C of width

D of width

60.0462.2

477.1 C of hypotenuse

D of hypotenuse

C D

Triangle D is 0.6 the size of triangle Canyway you look at it:

Ratios of any two sides within one figure are in the same proportion

as the corresponding ratio of any similar figure.

1 inch

2.25

inch

1.35

inch

0.6 in

2.462 inch

1.477 inch

25.2width

height25.2

6.0

35.1

094.1height

hypotenuse094.1

35.1

477.1

462.2width

hypotenuse462.2

60.0

477.1

You may also remember

C D

Each triangle is 2.25 as tall as it stands wide.

A description that applies equally to each.

Quantities are in proportion when theysimply 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 volumedensity (of water)

The charge built up in a capacitor scaleswith 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

Cos

t (i

n do

llar

s)

Gallons (of gasoline)

$30

$25

$20

$15

$10

$ 5

5 10 15

cost = $2.34/gal number of gallons

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

F = C + 3295

Not all relationships are proportions:

Temperature conversion follows:

oF

oC

y = mx + b

also: not all relationships are even linear!

Notice: the graph of F vs C does not go through (0,0)!

This means while 0o C = 32o Fdoubling (or tripling) both gives0o C and 64o (or 96o) F which is

no kind of proportion!

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” anglesof every triangle add up to 360o…

A

B

C

D

EF

A little geometry:

…these two triangles are similar!

Look carefully to see which are the“corresponding” (matching) sides.

Proportionalities

AC

?

1.E/F2.F/E3.D/F4.F/D5.E/D6.D/E

A

B

C

D

EF

The ratio of lengths of side A to side Cis in the same proportion as:

ProportionalitiesA

B CD

wx

A z

?

1.A/w2.A/x3.B/x4.C/x5.D/x6.D/w

y

z

R

A a

?

1.R/r2

2. R2/r2

3. R2/r4. R/r5. r/R

rThe ratio of surface areas

A

a

S

s

height, h

weight, W

Consider this block weighing “W”

height, 2hThis stack of 2 blocks

weighs how much?

2W

Are these blocksin proportion?

To scale proportionally

height, h

weight, W

And this double-sized block weighs

1. 2W 3. 6W 5. 10W2. 4W 4. 8W 6. 12W

More generally,

h

wL 2h

2w2L

originalvolume =hwL

newvolume

=(2h)(2w)(2L)

= ( 8 )hwL

= ( 8 )( )original volume

Is weight meant to be proportional to height?

Weight (Height)3

Each 1% increase in height shouldcorrespond 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

5. C and GQuestion 3“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).

We already demonstrated in class that I lifting requires,on average, a force simply equal to a load’s weight.

2. It lifts with a force = the load’s weight.

Question 1

A. Lifting box up from floor. requires positive workB. Holding box above floor. requires NO workC. 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 2 A only!

A, the shortest side of the larger triangle corresponds to F, the shortest of the 2nd triangle. C is the middle-sized side so corresponds to D.

Question 4 4. F/D

5. D/xQuestion 5A and z are bases of their respective (isosceles) triangles, i.e, they are “corresponding sides.” D is an “altitude” and so corresponds to x.

2. R2/r2Question 6The 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.

Question 7 4. 8WTwice as wide…twice as tall…twice as thick…means 222=8 times the volume (and mass).