PHYS 9B Lecture

What is the buoyant force?

The upward force exerted by a fluid on an object that is partially or completely submerged.

What causes the buoyant force?

Pressure is greater at the bottom of an object than at the top, producing a net upward force.

State Archimedes' principle.

The buoyant force on an object equals the weight of the fluid displaced by the object.

What is the equation for buoyant force?

FB = ρfluid gVsub.

What does ρfluid represent in the buoyant-force equation?

The density of the displaced fluid, not necessarily the density of the object.

What does Vsub represent in the buoyant-force equation?

The submerged volume of the object, which is also the volume of fluid displaced.

What volume is used when an object is completely submerged?

Vsub = Vobject.

What volume is used when an object is only partially submerged?

Only the portion of the object's volume below the fluid surface.

In what direction does the buoyant force act?

Upward.

What forces act on an object floating freely at rest?

An upward buoyant force and a downward gravitational force.

What condition applies to an object floating at rest?

FB = Fg = mg because the net force is zero.

What determines whether an object sinks or floats?

Its average density compared with the density of the surrounding fluid.

What happens if ρobject is greater than ρfluid?

The object sinks unless another force supports it.

What happens if ρobject is less than ρfluid?

The object floats with only part of its volume submerged.

What happens if ρobject equals ρfluid?

The object can remain completely submerged with neutral buoyancy.

What is the relationship between density and submerged fraction for a floating object?

Vsub/Vobject = ρobject/ρfluid.

A ball floats with 25% of its volume submerged in water. What is its density?

ρobject = 0.25ρwater = 250 kg/m³.

What is apparent weight?

The supporting or normal force measured while an object is submerged.

What is the equation for apparent weight?

Apparent weight = true weight - buoyant force.

An object rests on the bottom of a fluid. What is its force-balance equation?

FN = Fg - FB.

Why does an object appear lighter underwater?

The upward buoyant force reduces the supporting normal force.

How can an iron boat float even though iron is denser than water?

Its hollow shape increases its total volume and allows it to displace enough water for buoyancy to equal its weight.

What happens when more weight is added to a floating boat?

The boat sinks farther, displacing more water and producing a greater buoyant force.

For any object floating at equilibrium, how much fluid weight is displaced?

A weight of fluid exactly equal to the total weight of the floating object.

A 200 g block and a 150 g iron mass float together. What water volume is initially displaced?

350 cm³ because 350 g of floating mass must displace 350 g, or 350 cm³, of water.

What is the volume of a 150 g iron mass if ρiron = 7.9 g/cm³?

V = m/ρ = 150/7.9 ≈ 19 cm³.

A 150 g iron mass sinks while a 200 g wooden block floats. What total water volume do they displace?

The block displaces 200 cm³ and the iron displaces about 19 cm³, for approximately 220 cm³ total.

What properties define an ideal fluid?

It is incompressible, has steady flow, and has no internal friction or viscosity.

What is steady fluid flow?

The velocity of the fluid at each location does not change with time.

What is the volume flow rate?

The volume of fluid passing a point per unit time.

What is the equation for volume flow rate?

Q = ΔV/Δt = Av.

What is the SI unit of volume flow rate?

Cubic meters per second (m³/s).

What does the continuity equation state?

A₁v₁ = A₂v₂ for steady flow of an incompressible fluid.

What quantity is conserved in the continuity equation?

Volume flow rate: Q = Av remains constant.

How does fluid speed change when a pipe becomes narrower?

Speed increases because the same volume must pass through a smaller area each second.

How does fluid speed change when a pipe becomes wider?

Speed decreases because the same volume can pass through the larger area at a lower speed.

How is pipe area related to its radius?

A = πr².

If a pipe's radius doubles, how does its cross-sectional area change?

The area becomes four times larger.

If a pipe's radius doubles, how does the fluid speed change?

The speed becomes one-fourth as large, assuming constant volume flow rate.

A 1,000 L tank fills in 120 s. What is the volume flow rate?

Q = 8.33 L/s = 8.33 × 10⁻³ m³/s.

Gasoline flows through an 8.0 cm diameter pipe at Q = 8.33 × 10⁻³ m³/s. What is its speed?

v = Q/(πr²) = 1.66 m/s.

The pipe diameter increases from 8.0 cm to 16 cm. What happens to the gasoline speed?

The area becomes four times larger, so the speed decreases from 1.66 m/s to about 0.41 m/s.

How can pressure be interpreted as an energy density?

P = F/A, and work = FΔx; therefore, pressure represents energy per unit volume.

What forms of energy density appear in Bernoulli's equation?

Pressure energy density p, kinetic energy density ½ρv², and gravitational energy density ρgh.

State Bernoulli's equation.

p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂.

What does Bernoulli's equation conserve?

The total mechanical energy per unit volume of an ideal fluid.

Under what conditions can Bernoulli's equation be used?

The fluid must be ideal, with steady, incompressible flow and negligible internal friction.

At equal elevations, how are fluid speed and pressure related?

Faster-moving fluid has lower pressure; slower-moving fluid has higher pressure.

If fluid speed remains constant but elevation increases, what happens to pressure?

Pressure decreases because gravitational energy density increases.

If a horizontal pipe narrows, what happens to speed and pressure?

Speed increases and pressure decreases.

Why should the continuity equation often be used before Bernoulli's equation?

The continuity equation can determine an unknown speed needed in Bernoulli's equation.

In the lecture pipe example, how is the lower gauge pressure converted to absolute pressure?

p₁ = 75 kPa + 101.3 kPa = 176.3 kPa.

In the lecture pipe example, what speed is found in the upper narrow pipe?

Using A₁v₁ = A₂v₂ gives v₂ = 11.25 m/s.

In the lecture pipe example, what is the final upper gauge pressure?

The absolute pressure is 105.9 kPa, so the gauge pressure is 105.9 - 101.3 = 4.6 kPa.

Why does pressure decrease in the upper narrow section of the lecture pipe?

The fluid moves faster and gains elevation, so both changes reduce its pressure.

A diving bell becomes three-fourths full of water. What happens to the trapped air?

Its volume becomes one-fourth of its original volume.

What is the absolute pressure of air compressed to one-fourth its original volume at constant temperature?

Using P₁V₁ = P₂V₂, the pressure becomes 4 atm, or approximately 404,000 Pa.

At the air-water interface inside the diving bell, how does water pressure compare with air pressure?

The pressures are equal at the interface, so the water pressure is also 4 atm.

How deep below sea level is water at 4 atm absolute pressure?

About 30 m below the surface because the gauge pressure is 3 atm.

In the diving-bell example, the air-water interface is 3 m above the bottom. What is the bottom depth d?

d ≈ 30 m + 3 m = 33 m.

What is a common mistake in buoyancy problems?

Using the object's full volume when it is only partially submerged.

What is a common mistake in continuity problems?

Using diameter directly instead of calculating area with A = πr².

What is a common mistake in Bernoulli problems involving gauge pressure?

Mixing gauge and absolute pressures; all pressure terms must use the same reference.