332 Physics –

Scaling & Fluids

Is King Kong fictional or could a gorilla grow to that enormous size?

Robert Wadlow was measured at 8ft 11in tall, and weighed 485 pounds at time of death in 1940.

Why aren’t humans growing to be 9ft or taller?

What happens to things when they are scaled up in size? We will look at area and volume.

How does the area of a cube grow if the cube is increased in size in all dimensions?

1cm 1cm

1cm

What is the total surface area for the entire cube?

What is the volume of the cube?

2cm2cm

2cm

What is the total surface area for the entire cube now?

What is the new volume of the cube?

We doubled each dimension. How does area and volume grow when dimensions are increased. What is the relationship?

It turns out that the S.A.

The volume

Square–Cube Law (Galileo)

EXAMPLE: Scale up an average human up in all dimensions…10x larger.

So, mythical giants…possible or not possible to exist? WHY?

b) By what factor does their volume increase?

a) By what factor does their S.A. increase?

c) How much heavier would they be?

The bones and muscles of the larger animals tend to be disproportionally thicker and larger than the smaller animals. Smaller animals will have the greater relative strength that allows them to jump higher and fall greater distances without harm.

This does NOT mean that an ANT is stronger than an ELEPHANT. It means that the ant can lift more per pound of bodyweight.

As insects scales up, this limits their ability to breathe which limits the size of insects.

Almost all life is made up of cells which have quite similar oxygen requirements. As larger creatures evolved (assuming shape basically stayed the same), the surface area decreased relative to the volume, so it became more difficult to absorb enough oxygen.

Physics and Evolution of animals

The evolutionary breakthrough to larger size animals came with

The intestine for larger animals to asbsorb food As animals got larger, they needed to be able to absorb nutrients. This problem is handled by replacing the straight tube gut (like in a worm) by one with many convolutions, in which also the smooth surface is replaced by one with many tiny foldsAfter your food leaves your stomach, it travels into a long tube called the small intestine. The small intestine is about 20-feet long, but it has many folds in it. This allows the nutrients to be absorbed through the small intestine’s walls & into your blood stream. Your blood stream takes the nutrients to all parts of your body to feed its cells.

Cells are the basic building blocks for life. Why then, do cells divide? Why not have a few large cells?

If two objects are heated to the same temperature (billiard ball vs bowling ball) and then allowed to cool, which object will cool off faster?

Heating and Cooling

Which is better for keeping warm?

Which is better for chilling a drink?

Which has more skin, a lb of grapes or a lb of grapefruit?

For the smaller mammals, a higher surface area to weight ratio, the constant consumption of food to maintain the required elevated body temperature is a primary concern of survival.

Mice, Bats, and Humming birds have a high rate of heat loss relative to their total body weight. None of these animals exists in extreme cold habitats. To replace the thermal energy that they lose so quickly these small animals are committed to a life of being constantly on the move for the purpose of finding and consuming as much food as possible. Their daily intake of food can be more than one third of the animal’s total weight. Humans only consume a couple of lbs a day.

Warm-blooded animals: Maintaining a constant body temperature

At the other end of the spectrum, elephants have a much lower surface-to-mass ratio so they have the opposite sizing problem in that they need a way to remove excess heat from their interior.

The evolutionary solution is large earflaps whose purpose is more geared towards the radiation of heat rather than improvements in hearing. Elephants pump warm blood to the skin of their earflaps where through convection and radiation the excess heat is removed.

Notice that the shape of the mouse is as close as it can be to that of a sphere, a shape that produces the minimum amount of surface area. Now notice that the elephant’s skin is covered with

Also notice that the mouse is like nearly all mammals in being covered with a thick fur coat while the elephant is nearly void of all hair.

SPHERES have the least S.A. per unit volume of any shape

Due to its low surface area to mass ratio, turkey requires more time for the heat to penetrate through to the interior. In contrast to a thanksgiving turkey that requires three to four hours in the oven, a pizza has a large surface area to mass ratio and so it only needs about 14 minutes to cook.

Cooking

Which cooks faster? Why?

Density

The density, ρ, of an object is its mass per unit volume:

The unit for density is kg/m3

Material Density (kg /m3)Aluminium 2700Gold 19300Ice 917Water 1000Oil 800Air 1.29Helium 0.179

PressurePressure is force acting perpendicular to a surface divided by the area A over which the force acts.

Unit of pressure: 1 N/m2 = 1 Pascal or Pa

Knives, scissors, needles, papercuts…Why are these items effective in cutting or puncturing?

Snowshoes, walking on broken glass

Pressure in Fluids

Pressure is the same in every direction in a fluid at a given depth; if it were not, the fluid would flow….static equilibrium.

How is pressure defined within a fluid?

Consider a beaker of water of area A and height, h.

h

The force exerted on top of water surface will be smaller than the force at bottom…why?

Ftop

Fbot

Determine the weight of the water on top of bottom surface.

In which container would a fish experience more pressure from the liquid if the fish was at the bottom of each container?

A B C D

Example 1

In raiders of the lost ark, Indiana Jones replaces a golden idol with a bag of sand to keep the same force on the pedestal. If the idols mass was 11.3kg, calculate the volume of the idol if it was solid gold (19,300kg/m3).

b) The density of sand is 1220kg/m3. If the sand was the same mass as the gold, how much space would it occupy? Comment on this. 1cc = 1x10-6 m3)

a) Determine the water pressure at the bottom.

b) Determine the force from the water on the bottom of pool.

Example 2: A cylindrical swimming pool with a diameter of 5.4m is filled to a height of 1.22m.

Why doesn’t a giraffe’s brain explode when it drinks from a puddle or faint when it raises its head to such a large height above the heart? Pressure is height dependent above or below the heart.

Explain the different ranges of water.

Pressure / Density Questions

1. Why do swimmers float better in the ocean than in a lake?

2. Why don’t they make longer snorkels so that people could dive deeper without scuba gear?

4. When ice cubes melt when floating in water, does the water level rise, fall, or remain the same?

3. Submarines change depth by taking/releasing water. Why does that affect depth?

Atmospheric Pressure

At sea level the atmospheric pressure is about 101.3kPa. This is called one atmosphere (1 atm). We also know this as 14.7 lb/in2.

air30km

What is force on piece of paper sitting on table?

Why doesn’t paper fall through table?

Atmospheric Pressure and Gauge Pressure

Most pressure gauges measure the pressure above the atmospheric pressure – this is called the gauge pressure.

A car tire is inflated to 35psi. Is that the gauge pressure?

The absolute pressure is the sum of the atmospheric pressure and the gauge pressure.

Each of the four tires of a 1450-kg car have a gauge pressure of 221,000Pa. How much area of each tire is in contact with the ground?

How much weight does each tire support?

If a problem asks for total force or total pressure, you have to factor in the pressure from the atmosphere

Barometer Barometers are used to measure atmospheric pressure. The height of the column in the tube is such that the pressure in the tube at the surface level is Patm.

Juice box & suction cups

To what length could water be drawn up a vertical straw?

Pressure & BoilingThe lower the pressure on a liquid, the easier it is to make it boil, i.e., as pressure increases, so does the boiling pt.

Buoyant Force (FB)The pressure at the top surface is given by:

The pressure at the bottom surface is given by:

Since P2 > P1 , there is a NET FORCE upward on the object due to the liquid. This is called Buoyant Force.

This is what makes objects feel lighter in the pool

Archimedes’ PrincipleAny fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces:

If the FB > Fg , then the object will rise upwards (out of liquid) until FB = Fg and object then floats.

If the Fg > FB (max), then the object will sink.

Archimedes’ Principle & Density

If ρobject > ρfluid then object will sink.

If ρobject < ρfluid then object will float.

If ρobject = ρfluid then object can be anywhere in fluid (fish).

1 liter of water weighs 9.8N

If you displace 1 liter of water, the buoyant force is 9.8N

A rock weighs 15N in air when hanging from a scale. It is lowered into a beaker of water and fully submerged. The scale then reads 13.4N.

a) Find the volume of the rock

b) Find the buoyant force on the rock

c) Find the density of the rock

For a floating object, the fraction that is submerged is given by the ratio of the object’s density to that of the fluid.

For example, ice is 90% the density of water. Therefore, a block of ice will float in water with 90% of it submerged.

Fraction submerged x density of fluid = density of object

Archimedes’ Principle for Gases

This principle also works in the air; this is why hot-air and helium balloons rise.

We have a buoyant force on us from the atmosphere.

Consider a boat loaded with scrap iron in a swimming pool. If the iron is thrown overboard into the pool, will the water level at the edge of the pool rise, fall, or remain unchanged?

Pascal’s PrincipleAny change in pressure applied to a completely enclosed fluid is transmitted equally to all parts of the fluid.

Brakes, hydraulics, syringes, etc

In a hydraulic lift, the diameter of the pistons are 5.0 cm and 20.0 cm. A car weighing 10,000N is to be lifted by the large piston.

Determine the amount of force necessary to lift the car.

Fluids in Motion

Equation of continuity

The volume flow rate (Av) has the same value at every point along a tube that has a single entry and exit point for the fluid flow. For two positions…

Bernoulli’s PrinciplePressure in a MOVING fluid is given by:

As fluid speeds up, pressure in fluid drops

P = absolute pressure

When a problem asks for gauge pressure, it wants the Patm subtracted out.

When calculating P1 or P2 this is the ABSOLUTE pressure. To find GAUGE pressure, you need to subtract away Patm.

Pabs = Patm + Pgauge

Note on Bernoulli equation

Application: Air moves faster over top of wing, slower beneath. This leads to lower pressure over wing and is one of factors that contributes to uplift of plane.

Example: What is the lift (net force) on a wing with an area of 75.0m2 if air passes over the top and bottom surfaces of the wing at speeds of 340 m/s and 305 m/s, respectively?

Example 2: Water at a pressure of 3xPatm at street level enters an office building at a speed of 0.650 m/s through a pipe with a radius of 5.08cm and proceeds up to the top floor 27.5 m above via a pipe that tapers down to 2.54cm in radius as it reaches the top floor.

A) Determine the speed of the water in the top pipe.

B) Determine the flow rate of the water.

C) Determine the pressure, P2, in the top pipe.

Application: Curve ball, air on one side of spinning ball faster than the other (leads to pressure difference)

Application: Atomizers, spray bottles, lawn sprayers

Fast moving area over top of tube creates low pressure at top of tube. Patm in bottle pushes liquid up into tube and emerges into fast moving air.