Water pressure on the bottom surface is the total weight of the water above it

A

Mg

A

gV )(ρ=A

gAh)(ρ= hgρ=

Plus the atmospheric pressure weighing down from above on the whole thing!

€

Patm + ρhg

However this pressure is not just directedDOWNWARD

€

+ρhgPatm

But outward as well

3.0 meters

Swimming along the 10-foot bottom

of the poolyour back supports a ~3 meter column

of water.

With cross-sectional area

A 0.75 m2

That’s a 3 0.75 = 2.25 m3

volume of water

with a weight of ρVg

)/8.9)(25.2)(000,1( 233 smm

m

kg=

= 22,050 Newtons

5000 pounds!

Water pressure at any level is Patm + the additional weight of the water above it

€

Patm + ρhg

Simple evidence of pressure increasing with depth:

Water flows throughthe upper horizontalsection of pipe with

a speed v

It must flow through thislower section of pipe with

A. a speed greater than v.

B. a speed equal to v.

C. a speed less than v.

Water flows throughthe upper horizontalsection of pipe with

a speed v

The pressure experienced by the waterin the lower section of pipe must be___ the pressure in the top section.

A. greater than

B. equal to

C. less than

Water flows continuously through thepoint where two sections of pipe join.

The pipe at left has diameter D. The pipe at right diameter d = D/2.

If water flows through the left pipe with a speed V it travels in the

right pipe with a speed of

A.

B.

C. V€

V4

€

V2

D. 2V

E. 4V

Water flows continuously through thepoint where two sections of pipe join.

The pipe at left has diameter D. The pipe at right diameter d = D/2.

The pressure experienced by water in the right pipe must be ___

the water pressure in the left pipe.

A. less than

B. equal to

C. greater than

Bernoulli’s Equation

P + ρv2 + ρgh = constant 12

When a stream of water either speeds up or flows uphill

the pressure it exerts drops!

When a stream of water slows or flows downhill

the pressure it exerts drops!

Speeding up or rising uses up energy!

A

B

C

D

Low pressure water from reservoir

Pumping station/purification plant

Water storage at low pressure

Low pressure water from reservoir

air at atmospheric

pressure

For

ce

Displacement, x

F x

Also means: F = kx

“spring constant” in units of N/m

The restoring force grows so many Newtons for every meter stretched.

Look up any catalog of springs.

Their “strength” or “spring rate” is given as “the load per unit deflection”

lbs/in or lbs/ft or N/mm or N/cm or N/m

9.5 lbs/in32 lbs/in45 lbs/in96 lbs/in

Springs ofincreasingstrength

or stiffness

For

ce

Displacement, x

Work done against an elastic force

starting with force 0

ending with force = kx

W = Favg d

= xFmax

2

= k(x)212

Dropped with

potential energymg(h1)

…equal tothe kinetic

energy itstrikes

ground with

Golf Ball

h1

rebounding with a slightly smaller kinetic energy

equal to the potential energymg(h2) it will climb up to.

h2

outgoing kinetic energyincoming kinetic energy

1

2

1

2

h

h

mgh

mgh==

This is a ratio of energy after

energy before

2

121

2

221

1

2

mv

mv

h

h=

Golf balls (and many industrial materials)are rated on their “coefficient of restitution”

outgoing speedincoming speed

Since our ratio: 2

1

2

2

v

v=

2

1

2

⎟⎟⎠

⎞⎜⎜⎝

⎛=

v

v

outgoing speedincoming speed

1

2

h

h=

An elastic ball wastes 30% of the collision energy as heat each time it bounces on a hard floor.

This ball will rebound with ___ thekinetic energy it strikes the floor with.

A. 0.70 C. E. (0.70)2 = 0.49B. 0.30 D. F. (0.30)2 = 0.09

€

0.70 = 0.84

This ball will rebound with ___ the speed it strikes the floor with.€

0.30 = 0.55

Liveliness of bouncing objectscan be described by either ratio:

ratios ofheights

ratios ofspeeds

Although they are related, theyare not the same number!

coefficient ofrestitution

Turning right on level ground relies entirely on friction

A banked curve let’s the car’s own weight help negotiate the turn

Note: the normal force is greater.

Note: the frictional forces are smaller.

Why?

Curves are usually banked so that you still make the turn even if friction vanishes!

N

f

The total surface forces are not off-centered at allso produce no net torque of their own!

f

N

At the bottom of the swing, the bucket must be supported by a force

A. less than its weight.B. equal to its weight.C. greater than its weight.

This wheel is rolling away from you (into the screen):

Its angular momentum

points in what direction?

A. right. B. left.C. down. D. up.E. into the screen.F. out of the screen.

Torques as well as angular momentaare both assigned directions by the

“right-hand rule.”

The approximately circular orbits of planetsand moons lead Newton to

221

R

mmF

grav∝

221

R

mmGF

grav=

G is a universal constant measured to be6.67 10-11 N·m2/kg2

0.000 000 000 066 7 N·m2/kg2

and for any moon orbiting any planet:

2

24

T

Ra

π=

For any planet (or star or asteroid)

2R

MGa planet=

the acceleration down toward its center is

which means:

2

2

2

4

T

R

R

MG planet π=

2

324

GT

RM

planet

π=

p = 0 pgas procket

pi = 0 = pf pgas = – procket= pgas + procket

pi = 0 = pf = prifle + pbullet prifle = – pbullet

The 90-kg fullback runs up the middle

at 8 m/secstraight into the waiting arms of

the 110-kgdefensive tackle.

The tackle wraps his arms about the fullback and the two are carried along

together by the fullback’s momentum with what speed?