A fluid is a substance (a liquid or a gas) that deforms continuously under an applied shear stress. In this unit, we focus on two key properties of liquids: surface tension and viscosity.
Surface Tension is the tendency of liquid surfaces to shrink into the minimum possible surface area. It is caused by cohesive forces (attraction between like molecules) being stronger at the surface than in the bulk liquid.
It can be defined in two ways:
Formula: T = F / L (Units: N/m)
Formula: T = W / ΔA (Units: J/m²)
Because of surface tension, a curved liquid surface (like in a drop or bubble) creates a pressure difference (ΔP) between the inside and the outside. The pressure is always higher on the concave side.
A drop has only one surface. The excess pressure (ΔP) inside is:
Formula: ΔP = Pinside - Poutside = 2T / r
Where T is the surface tension and r is the radius of the drop.
A soap bubble has two surfaces (an inner and an outer surface) in contact with the air. Both contribute to the pressure.
Formula: ΔP = Pinside - Poutside = 4T / r
For a long cylindrical drop of radius r (like a dewdrop on a thin wire), which has one curved surface:
Formula: ΔP = T / r
Note: A cylindrical bubble (like a long bubble in a liquid) would have ΔP = 2T/r.
Surface tension is a result of cohesive forces. As temperature increases, the kinetic energy of the molecules increases, and the cohesive forces weaken.
Therefore, the surface tension of a liquid decreases as temperature increases.
At the critical temperature, the distinction between liquid and gas disappears, and the surface tension becomes zero.
Viscosity is the measure of a fluid's resistance to flow. It's the "internal friction" of a fluid. High viscosity (like honey) means high resistance to flow; low viscosity (like water) means low resistance.
Newton's law of viscosity states that the shear stress (F/A) required to maintain a velocity gradient (dv/dz) between fluid layers is:
Formula: F/A = η * (dv/dz)
Where η is the coefficient of viscosity.
When a fluid flows steadily (laminar flow) through a narrow tube (capillary) of radius r and length L, driven by a pressure difference ΔP, the volume flow rate (Q) (volume per second) is given by Poiseuille's Formula:
Poiseuille's Formula: Q = V/t = (π * ΔP * r⁴) / (8ηL)
For liquids, viscosity is primarily due to intermolecular cohesive forces.
(Note: This is the opposite for gases, where viscosity *increases* with temperature, as it's caused by molecular collisions, not cohesion.)
The Special Theory of Relativity (STR), proposed by Albert Einstein in 1905, describes physics in inertial frames of reference (frames that are not accelerating).
An inertial frame is one where Newton's First Law (inertia) holds. Any frame moving at a constant velocity relative to an inertial frame is also an inertial frame.
A Galilean Transformation is the "common sense" way to relate coordinates between two inertial frames (S and S') where S' moves at a constant velocity v relative to S (e.g., along the x-axis).
Galilean Transformation:
- x' = x - vt
- y' = y
- z' = z
- t' = t (The "common sense" part: time is absolute)
Problem: This transformation works for mechanics, but it fails for electromagnetism. Maxwell's equations predict a constant speed of light, c. According to Galileo, an observer in S' should measure c' = c - v, but experiments (like Michelson-Morley) showed the speed of light is *always* c for all observers.
Einstein fixed this by proposing two new postulates:
Postulate 1 (The Principle of Relativity): The laws of physics are the same (have the same form) in all inertial frames of reference.
Postulate 2 (The Constancy of the Speed of Light): The speed of light in a vacuum (c) has the same value for all observers in all inertial frames, regardless of the motion of the source or the observer.
To make both postulates true, the Galilean transformations must be wrong. The new, correct transformations are the Lorentz Transformations. We must find a transformation that keeps the speed of light constant.
Derivation Outline:
x' = γ (x - vt) (We introduce a "fudge factor" γ)
x = γ' (x' + vt') (The inverse, from S' to S)
x = γ (x' + vt')
ct' = γ (ct - vt) = γt (c - v)
ct = γ (ct' + vt') = γt' (c + v)
(ct') (ct) = [γt (c - v)] [γt' (c + v)]
c² t' t = γ² t t' (c - v)(c + v)
c² = γ² (c² - v²)
γ² = c² / (c² - v²) = 1 / ( (c² - v²) / c² ) = 1 / (1 - v²/c²)
γ = 1 / √(1 - v²/c²) (This is the Lorentz factor)
t' = γ (t - vx/c²)
Lorentz Transformations:
- x' = γ (x - vt)
- y' = y
- z' = z
- t' = γ (t - vx/c²)
where γ = 1 / √(1 - v²/c²)
Note: If v << c, then v²/c² ≈ 0, γ ≈ 1, and vx/c² ≈ 0. The Lorentz transformations simplify to the Galilean transformations, which is why we don't notice relativity in everyday life.
A consequence of the Lorentz transformations. An object of length L₀ (its "proper length") in its own rest frame will appear shorter to an observer moving relative to it.
The measured length L is:
Formula: L = L₀ / γ = L₀ * √(1 - v²/c²)
Result: Moving objects appear shorter in the direction of their motion.
Another consequence. Time passes slower for moving clocks.
If an observer measures a time interval T₀ (the "proper time") on a clock at rest with them, an observer moving at speed v will measure a *longer* time interval (T) for that same event.
Formula: T = γ * T₀ = T₀ / √(1 - v²/c²)
Result: Moving clocks run slow.