## I. 6.1: HYDROSTATICS: FLUIDS AT REST

### A. FLUIDS

1. Fluids – substances that can flow (both liquids and gases)

### B. DENSITY AND SPECIFIC GRAVITY

1. Density – the amount of substance in unit volume

a) ρ = m/V

b) Specific gravity = (density of substance)/(density of water) → no units!!

(1) Density of gas changes markedly with pressure and temperature (not the case with liquids)

### C. FORCE OF GRAVITY FOR FLUIDS

1. ρ = m/V → m = ρV → ∴ mg = **F _{grav} = ρgV**

### D. PRESSURE

1. P = F⊥/A = (force⊥)/area

a) ⊥ indicates that the force is perpendicular to the surface

b) Force can be the weight of a fluid

2. Pressure is in N/m^{2} = Pascal (Pa); 1 ATM ≈ 100 kPa

3. **P _{guage} = ρ_{fluid}gD** → hydrostatic guage pressure; does not include atmospheric pressure on top

a) D = depth of fluid; V = A*depth

4. P_{total} = P_{atm} + P_{guage}

### E. BUOYANCY AND ARCHIMEDES’ PRINCIPLE

1. **Buoyant force** – upward fluid force on something that is floating (forces on the sides of the object cancel out)

2. **Archimedes’ Principle**: the magnitude of the buoyant force is equal to the weight of the fluid displaced by the object

a) **F _{buoy} = ρ_{fluid}V_{sub}g** → multiply the volume of fluid displaced x density, you get mass

b) **Floating objects**:

(1) **w _{object} = F_{buoy}**

(2) **V _{sub}/V = ρ_{object}/ρ_{fluid}** → will tell you percentage of object is submerged

(a) The fraction of the volume of the object submerged is the same as the ratio of its density to the fluid’s density

c) ** Sinking objects**:

(1) The object’s weight is > F_{buoy}

(2) Since volume of object = submerged volume, then the **F _{buoy} = ρ_{fluid}V_{sub}g**

(3) If the fluid in which the object is submerged is water, the ratio of the object weight to the boyant force is equal to the specific gravity of the object

(a) **w _{object}/F_{Buoy} = ρ_{obj}/ρ_{fluid}**

### F. PASCAL’S LAW

A fluid will transmit an externally applied pressure change to all parts of the fluid and the walls of the container with loss of magnitude (think of hydraulics → you apply pressure at one end and that pressure is transmitted through to the other side)

1. **F _{1}/A_{1} = F_{2}/A_{2}** → F

_{2}= (A

_{2}/A

_{1})F

_{1}

2. d_{2} = (A_{1}/A_{2})d_{1}

### G. SURFACE TENSION

## II. 6.2: HYDRODYNAMICS: FLUIDS IN MOTION

### A. FLOW RATE AND THE CONTINUITY EQUATION

1. Flow rate (f) – volume of fluid that passes through a particular point per unit time (m3/sec)

a) f(or Q) = Av → A is cross sectional area of pipe

b) Don’t confuse flow rate with flow speed! *(m ^{3}/sec vs m/s)*

c) Flow rate must be the same everywhere along the pipe because fluid is incompressible

2. Continuity equation – if flow rate is the same everywhere in the pipe, then the A*v must be the same

a) **A _{1}v_{1} = A_{2}v_{2}** → this means the flow speed is inversely proportional to the cross-sectional area

### B. BERNOULLI’S EQUATION

Most important equation in fluid dynamics!

1. Applies to ideal fluids (must satisfy the following requirements)

a) Fluid is incompressible (doesn’t work well with gases)

b) There is negligible viscosity

c) Flow is laminar (not turbulent)

d) Flow rate is steady

2. If these conditions hold, then there is conservation of total mechanical energy for ideal fluid flow

3. **P _{1} + ½ρv_{1}^{2} + ρgy_{1} = P_{2} + ½ρv_{2}^{2} + ρgy_{2}**

a) ρ = density of flowing fluid

b) P_{1} and P_{2} give pressures at any two points along a streamline within the flow

c) v_{1} and v_{2} give glow speeds at these points

d) y_{1} and y_{2} give heights of these points above some chosen horizontal reference line

e) Bernoulli’s priniciple (y_{1} = y_{2}) ∴ **P _{1} + ½ρv_{1}^{2} = P_{2} + ½ρv_{2}^{2}**

4. Toricelli’s result:

a) **v _{efflux} = √2gD**

**→ v**

_{efflux}is speed

(1) D → distance from surface of liquid down to the hole

(2) Note that the v_{efflux} ∝**√D **, therefore v_{efflux}^{2} ∝ D

### C. THE BERNOULLI EFFECT → the P is ↓ where the flow speed is ↑

## III. 6.3: THE ELASTICITY OF SOLIDS

Examines the relationship between forces applied to a solid object and the resulting change in the object’s shape

### A. STRESS

1. 3 types:

a) Tension (stretching)

b) Compression (squeezing)

c) Shear (bending)

2. Stress = force/area = F/A → note that the force is not necessarily perpendicular to area; stress is inversely ∝ to the cross sectional area

### B. STRAIN

The ratio of the appropriate change in the length to the object’s original length

1. Tensile or compressive strain

a) strain = △L/L_{0} → the ratio of change in length to the original length

2. Shear strain

a) strain = X/L_{0} → the ratio of distance of shear (X) to original length

### C. HOOKE’S LAW

Stress causes strain; and assuming stress isn’t so large that it causes a permanent deformity (object is elastic), then stress and strain are proportional.

1. Young’s modulus – constant of proportionality for tensile/compressive stress (denoted Y or E)

a) Y or E = stress/strain = (F/A)/(△L/L_{0})

b) This is constant until things break!!

2. Shear modulus – constant of proportionality for shear stress (denoted S or G)

3. Hooke’s law

a) stress = modulus*strain → modulus = stress/strain

b) ∴ modulus and strain are inversely related

4. Tension/compression

a) F/A = E(△L/L_{0})

b) ∴ △L = FL_{0}/EA (Flea formula)

5. Shear

a) F/A = G(X/L_{0})

b) ∴ X = FL_{0}/AG (Flag formula)

6. Bulk modulus:

a) B = P/(△V/V) → not in book, but memorize

## CHAPTER 6 SUMMARY

- Assume liquids are incompressible unless otherwise stated
- Standard ATM = 1 atm = 760 mmHg = 100 kPa
- Density:
**ρ = m/V** - Specific gravity → ρ/ρ
_{H2O}(ρ_{H2O}= 1000kg/m^{3}or 1 g/cc or 1 kg/L) - Force of gravity →
**mg = ρVg** - Pressure →
**P = F**_{⊥}/A - Hydrostatic guage pressure →
**P**_{guage}= ρ_{fluid}gD- Hydrostatic guage pressure is ∝ depth; total pressure increases, but is
*not*∝ depth - Total hydrostatic pressure → P
_{total}= P_{at surface}+ P_{guage}

- Hydrostatic guage pressure is ∝ depth; total pressure increases, but is
- Archimedes principle →
**F**_{buoy}= ρ_{fluid}V_{sub}g- Byouant force is equal to weight of displaced fluid

- Floating object → if ρ
_{object}< ρ_{fluid}→ w_{object}= F_{buoy}**V**_{sub}/V = ρ_{object}/ρ_{fluid}

- Apparent weight of submerged object → w
_{apparent}= w_{object}– F_{buoy} - Pascal’s law →
**F**_{1}/A_{1}= F_{2}/A_{2} - Conditions ideal for fluid:
- incompressible
- negligible viscosity
- laminar flow
- steady flow

- Flow rate → f = Av
- Continuity equation → A
_{1}v_{1}= A_{2}v_{2} - Bernoulli’s equation
- Total energy (density) within all parts of an ideal fluid is the same
- P
_{1}+ ½ρv_{1}^{2}+ ρgy_{1}= P_{2}+ ½ρv_{2}^{2}+ ρgy_{2}

- Bernoulli’s priniciple
**P**_{1}+ ½ρv_{1}^{2}= P_{2}+ ½ρv_{2}^{2}- Fast flowing fluids have low pressures
- Slow flowing fluids have high pressures

- Toricelli’s result
**v**_{efflux}=

- Stress →
**stress = F/A** - Strain → strain =
**△L/L**(tension/compression) or_{0}**X/L**(shear)_{0} - Hooke’s law → stress = modulus*strain
- Tension/compression →
**△L = FL**_{0}/EA - Shear →
**X = FL**_{0}/AG