Density

1
Density Pressure

• Sometimes a force is not the concept that quite
  fills the bill...
• We saw this with work...
• Now consider a nail and a balloon...
• Beakmans World!

2
Pressure

• Pressure is a force / area
• The force is perpendicular to the area.
• The SI unit is the pascal (Pa).
• This is a very small unit.
• Usually, we will talk about kPa.
• Can you think of an example where pressure is a
  more useful concept than force?

3
Problem 1

• Does anyone dance ballet?
• What is the difference in pressure between
  standing flat-footed and on point?

4
Density

• Another useful concept is density.
• We need a way of comparing the size and mass of
  objects that is independent of what we happen to
  have at hand, right now.
• Compare a lead brick with clay brick
• Same massthe lead piece is much smaller.
• Same volumethe clay brick is much lighter.
• Density is the concept that we need.
• Density is the mass of an object divided by its
  volume.
• It is roughly the same for a given type of
  object, independent of how much you have.

5
Density

• The SI unit for density is kg/m3.
• There is no special name.
• 1 kg/m3 is a very small unit.

6
Pressure in Fluids

• We live our entire lives in fluids!
• This includes both liquids and gases.
• We will assume an ideal fluid.
• What else is new?!
• The force exerted by a fluid on a submerged
  object at any point is perpendicular to the
  surface of the object.
• This includes the sides of the container.

7
Pressure in Fluids

• The pressure at a given level in a fluid is the
  same.
• We assume the density is constant throughout the
  fluid.
• The pressure does not depend on the shape of the
  container.
• The pressure increases with depth.

8
Pressure in Fluids

• Pascals principle
• Pressure applied to an enclosed fluid is
  transmitted undiminished to every point of the
  fluid and to the walls of the containing vessel.

9
Problem 2

• The radius of the small piston is 5.00 cm,
  whereas the radius of the large piston is 30.0
  cm. If a force of 2.00 N is applied to the small
  piston, what force will occur at the large
  piston?
• 72.1 N

10
Problem 2a

• Suppose the large piston moves 0.200 cm. By how
  much must the small piston be moved?
• Use energy conservation.
• Work out Work in
• 7.21 cm

11
Problem 3

• (a) What is the total pressure on the back of a
  scuba diver in a lake at a depth of 8.00 m?
• (b) What is the force on the divers back due to
  the water alone taking the surface of the back to
  be a rectangle 60.0 cm by 50.0 cm?
• Assume normal atmospheric pressure.
• a) 179 kPa
• b) 23,500 N (5290 lb!)

12
Pressure Measurements

• Gauge pressure P - P0 rgh

13
Archimedes Principle

• Any body completely or partially submerged in a
  fluid is buoyed up by a force equal to the weight
  of the fluid displaced by the body.

14
Archimedes Principle
15
How dense is the crown?
16
Problem 4

• A cup is filled to the brim with water and
  floating ice cube. When the ice melts, which of
  the following occurs? (a) Water overflows the
  cup, (b) the water level decreases, or (c) the
  water level remains the same?
• (c)

17
Problem 5

• The density of lead is 11.3 X 103 kg/m3, so a
  solid piece of lead will sink. But a piece of
  lead shaped like a boat can float!
• Suppose you have a lead boat shaped like a
  rectangular box with no top. If 1000 kg of lead
  makes a box of length 2.0 m, width 2.0 m and
  height 30.0 cm, does it float? If so, how much of
  the boat is above water?
• 5 cm of the boat is above water - so yes, it
  floats.

18
Moving Fluids

• Assumptions
• The fluid is non-viscous.
• The fluid is incompressible.
• The fluid motion is steady.
• The fluid moves without turbulence.
• Equation of continuity

19
Problem 6

• To fill a childs inflatable wading pool you use
  a garden hose with a diameter of 2.8 cm. Water
  flows from this hose with a speed of 1.1 m/s. How
  long will it take to fill the pool to a depth of
  26 cm if it is circular and has a diameter of 2.0
  m?
• 1200 s 20 min

20
Bernoullis Equation

• Conservation of energy says that the total energy
  of the fluid at any point is constant.
• This energy is a combination of kinetic,
  potential energy and the work done by the
  pressure in moving the liquid.

21
Problem 7

• How do the pressures compare?
• Use both continuity equation and Bernoullis
  equation.
• Notice that h is the same for both.
• Faster fluids reduce the pressure!

22
A demonstration
23
Daniel Bernoulli

• 1700 - 1782