Fluids, Pressure and buoyancy

CAPA due Friday at 10pm. Comment on the hint in Problem 5.

CAPA solutions from previous sets can be found by logging onto CAPA and selecting “View Previous Set” instead of “Try Current Set”

Any special exam requests – should contact Daniel.Dessau@colorado.edu

Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/

Announcements:

Fluids A gas is a bunch of molecules flying about, occasionally colliding with other molecules or the walls of a container.

Because there is lots of space between molecules, a gas is compressible.

In a liquid, molecules are close together but not strongly bonded together so they can slide around. A liquid is incompressible since the molecules are already very close together.

Liquids and gases are both fluids. Fluids do not retain their shape. They also have the property of being able to flow.

Pressure You have encountered the concept of pressure.

Atmospheric pressure in weather reports Air pressure in car or bicycle tires

Pressure against eardrums in plane or underwater.

Pressure forces water out of the holes The fluid exerts a force against the walls of the container. Force is related to the pressure.

Define pressure:

SI unit is pascal: 1 pascal = 1 Pa = 1 N/m2 The force is perpendicular to the area

Pressure is a scalar

Pressure in a gas The gas molecules colliding with the walls exert a force on the walls.

An object inside also feels the force of the collisions (equally from all directions so net force is 0).

But it is not just the wall that feels the pressure. The object inside also feels the pressure even though the net force is 0.

The more area a wall has, the more collisions it will endure and the larger a force it will feel. So the force on the wall should be proportional to area: where p is the pressure (which is how we defined pressure!)

Atmospheric pressure (of air) at sea level is defined by a non-SI unit of atmosphere (atm). 1 atm = 101,300 Pa.

In Boulder our air pressure is about 80% of that at sea level

Pressure in a liquid Consider a column of water with depth d and cross sectional area A inside a container open to the atmosphere.

When you descend in a liquid, the weight of the liquid above you causes the pressure to increase.

Atmospheric pressure p0 pushes down with force of p0A. The weight of the column pushes down with force mg. For a liquid with density ρ, m=ρV=ρAd.

Because the liquid is in static equilibrium, the upward force from pressure, pA, must equal the downward forces

so (hydrostatic pressure)

Clicker question 1 Set frequency to BA

Three vessels are full of the same liquid and open to the same atmosphere. The pressure is measured in each at a distance of 3 m below the surface. What can we say about the pressures? A. only two are the same B. all three are different C. all three are the same 3 m

Clicker question 1 Set frequency to BA

Three vessels are full of the same liquid and open to the same atmosphere. The pressure is measured in each at a distance of 3 m below the surface. What can we say about the pressures? A. only two are the same B. all three are different C. all three are the same 3 m

The hydrostatic pressure is

It only depends on the pressure on top and the amount of water in a column directly overhead so it is the same for all 3.

Pressure in a liquid In the problem, the liquids had the same height because they were filled that way.

If they were all connected (as shown), the liquid levels would have to be the same. Assume the first container has a higher level. Then, since , pressure at A is greater than at B.

A B

Assume a gradual decrease from A to B. Then at any point between them, pressure from the left is greater than from the right resulting in a net force to the right (not equilibrium). Therefore, fluid will flow to the right until equilibrium is reached.

Some rules for pressure Anywhere in a connected, static, uniform density fluid, the pressure at a given height is the same. Pascal’s law: A pressure change at one point in an incompressible fluid is transmitted undiminished to all points in the fluid + walls. A force F is applied to a piston of area A, increasing the pressure by F/A.

Clicker question 2 Set frequency to BA

Uma Thurman (mass of 60 kg) is standing on a piston, connected as shown to another piston on which a 6000 kg stretch Hummer is resting. How much bigger in area is the piston under the Hummer compared to the one under Uma? Try writing down two expressions for the pressure at the dotted line. A. same B. 10 times C. 100 times D. 1000 times E. 10000 times

Both pressures at same height so must be the same. Setting equal:

Clicker question 2 Set frequency to BA

Uma Thurman (mass of 60 kg) is standing on a piston, connected as shown to another piston on which a 6000 kg stretch Hummer is resting. How much bigger in area is the piston under the Hummer compared to the one under Uma? Try writing down two expressions for the pressure at the dotted line. A. same B. 10 times C. 100 times D. 1000 times E. 10000 times

On the left side we have

Right side:

so so

Buoyancy Take a volume of water with density ρf and area A extending from a depth of d2 to d1. Summing the forces of the free body diagram gives

If we replace the fluid with an object, the only difference is mass.

Note that A(d1-d2) is the volume Vf of fluid displaced Upward force equals the weight of displaced fluid

Archimedes principle A body partially or fully immersed in a fluid feels an upward force equal to the weight of the displaced fluid.

This force is called the buoyant force:

As shown, it is due to the increase of pressure with depth in a fluid.

If the object is fully immersed then the volume of the displaced fluid is equal to the volume of the object:

Note that volume is related to mass and density:

If an object is only partially submerged, the volume of the displaced fluid is less than the volume of the object:

Buoyancy example A 2 cm by 2 cm by 2 cm cube of iron (ρ=8 g/cm3) is weighed with the iron outside, half in and fully in the water, as shown in the diagram. What is the measured weight in each case?

Iron mass: Out of the water:

so

In the water:

so

so

½ in the water:

so

so

Solving buoyancy problems Try to figure out the weight of the displaced fluid (buoyant force!)

If object is submerged, volumes of object and displaced fluid are equal

If object is floating, can use the fraction of the object that is submerged to relate the two volumes (object & displaced fluid).

With the displaced fluid volume, can use density and g to get weight.

If the object is in equilibrium or you know the acceleration, use Newton’s 2nd law.

If an object is floating (and no other forces are acting), buoyant force equals the weight of the object as well as the weight of the displaced fluid.

Clicker question 3 Set frequency to BA

A. larger. B. smaller. C. the same.

Two bricks are held under water in a bucket. One of the bricks is lower in the bucket than the other. Compared to the higher brick, the upward buoyant force on the lower brick is…

Clicker question 3 Set frequency to BA

A. larger. B. smaller. C. the same.

Two bricks are held under water in a bucket. One of the bricks is lower in the bucket than the other. Compared to the higher brick, the upward buoyant force on the lower brick is…

The weight of the displaced water is the same for both so the buoyant force is the same.

The pressures on the top and bottom of the lower brick are larger than the pressures on the top and bottom of the higher brick but the pressure differences are the same and this is the source of the buoyant force.