The pressure is on

Which is the best design for a dam? Explain your answer.

Which dam is more likely to break? Explain your answer.

Pressure Differential Whiteboard

A large cylindrical water tank is sealed with a vacuum on top, and develops a small hole at a depth h below the surface of the water, as shown above. Assuming that atmospheric pressure is 101,000 Pa and the density of water is 1,000 kg/m3, what is the maximum possible depth of the hole that will result in no water leaking out? 

(A) 10 m (B) 100 m (C) 1,000 m

(D) Water will leak out, no matter what.

(E) Water will not leak out, no matter what.

At the hole,

there is an outward force caused by the water above the hole

creating a large gauge pressure…

but there is also an inward force caused by the atmospheric

pressure!

When the outward gauge pressure (ρgh) from the weight of the fluid matches the inward force from

atmospheric pressure, the fluid will be in equilibrium.

Pressure depends only on depth and density of the fluid!

This means that if you have an irregularly shaped container of fluid, any two points with the same depth

will have the same pressure

This is the principle used in hydraulics.

P1 P2

P1 = P2

A hydraulic lift is a container of fluid with a large cross-sectional area on one end, and a small cross-sectional area on the other.

Because the pressure is the same at both ends (since they are at the same height), a hydraulic lift acts as a force multiplier!

P1 = P2 F1/A1 = F2/A2

The only tradeoff is that you will also have to push further in order to lift the larger end.

(Equal amounts of fluid must be displaced on either side)

Why does the liquid stay in the straw when you plug the end?

Before lifting it out of the glass, you need to press your finger tightly against the top of the straw.

This seals a finite amount of air in the top part of the straw, which is initially at atmospheric pressure.

However, when you lift the straw…

the water level drops!!!

Now the same amount of air is occupying a larger space inside the straw, and the pressure inside will be

lower than atmospheric pressure!

Less than atmospheric pressure

Atmospheric pressure

In order for the liquid to stay in the straw, it must be true that

Pinside + ρgh = Poutside

Gauge pressure from the weight of the fluid

The water level will continue to drop until this equation is satisfied

Did you realize…

that houses can float?that houses can float?

That entire cities can float?

Why does buoyancy exist?As you go deeper into a fluid, the pressure increases.

Pressure is isotropic! It pushes inward on all sides of a submerged object.

Differential PressureThe bottom of the object is at a greater depth than the top of the object.

The downward force exerted on the top of the object is less than the upward force exerted on the bottom of the object.

Buoyant pressure

hsubmerged

Ptop = Patm Pbottom = Patm + ρfluidghsubmerged

Pbuoyant = Pbottom – Ptop

Pbuoyant = ρfluidghsubmerged

ρfluid

A partially submerged block floating in a liquid.

Patm

Determining Buoyant ForceA

hsubmerged

ρfluid

Pbuoyant = ρfluidghsubmerged

Fbuoyant = Pbuoyant*A

P = F/A

Fbuoyant = ρfluidghsubmergedA

Determining Buoyant ForceA

hsubmerged

ρfluid

Fbuoyant = ρfluidghsubmergedA

Fbuoyant = ρfluidgVsubmerged

Vsubmerged

Fbuoyant = ρfluidgVsubmerged

1) The buoyant force is directly proportional to the density of the fluid.

- More dense fluids exert a greater buoyant force!

2) The buoyant force is also directly proportional to the amount of volume that is submerged.

- The buoyant force depends on the amount of liquid that is displaced by the object.

Rubber ducky, you’re the one.

A 10-g rubber ducky floats in a tub of water at bath time.

Draw and label a force diagram for the ducky.

How much of the volume of the ducky must be submerged in order for it to float?