Surface tension is a property of a liquid that allows it to resist an external force. It is the tendency of liquid surfaces to shrink into the minimum possible surface area.
Because of surface tension, the pressure *inside* a curved liquid surface (like a drop or bubble) is greater than the pressure outside. This is called excess pressure (ΔP).
A liquid drop has one surface (liquid-air).
The outward force from the excess pressure (ΔP × Area) must balance the inward "squeeze" from surface tension (T × Circumference).
A formal derivation (balancing work done) gives:
ΔP = Pin - Pout = 2T / R
Where `T` is surface tension and `R` is the radius of the drop.
A bubble (like a soap bubble in air) has two surfaces (an inner liquid-air surface and an outer liquid-air surface) that both contribute.
This effectively doubles the surface tension effect.
ΔP = Pin - Pout = 4T / R
For a cylindrical drop of liquid (like a long, thin stream), the curvature is only in one dimension (when viewed from the side).
ΔP = Pin - Pout = T / R
A cylindrical *bubble* (like a long, hollow film) would have two surfaces, giving `ΔP = 2T/R`.
Viscosity is the measure of a fluid's resistance to flow. It is essentially "internal friction" within the fluid.
When a fluid flows, different layers move at different speeds (this is called "shear"). The viscous force `F` between two layers is given by Newton's law of viscosity:
F = -η A (dv/dy)
Where `η` (eta) is the coefficient of viscosity, `A` is the area of the layer, and `dv/dy` is the velocity gradient (how fast the velocity changes with distance).
This is a key difference between liquids and gases.
This formula describes the volume flow rate (Q) of a viscous, incompressible fluid flowing in a laminar (smooth, non-turbulent) way through a cylindrical tube of length `L` and radius `R`.
The flow is driven by a pressure difference `ΔP = P₁ - P₂` between the ends of the tube.
The formula is:
Q = V / t = (π ΔP R⁴) / (8 η L)
In 1905, Albert Einstein proposed the Special Theory of Relativity to resolve conflicts between classical mechanics (Newton/Galileo) and electromagnetism (Maxwell). It is based on two fundamental postulates.
"The laws of physics are the same (invariant) in all inertial frames of reference."
This extends the (classical) Galilean principle of relativity (which only applied to mechanics) to *all* laws of physics, including electromagnetism. It means there is no "absolute rest frame" or "ether". Any inertial frame is as good as any other.
"The speed of light in a vacuum (c) has the same value (c ≈ 3 × 10⁸ m/s) for all inertial observers, regardless of the motion of the source or the observer."
This is a radical break from classical intuition.
Classical example: If you are on a train moving at 50 km/h and throw a ball forward at 20 km/h, someone on the ground sees the ball moving at 50 + 20 = 70 km/h.
Relativistic example: If you are on a spaceship moving at 0.9c and turn on a headlight, you measure the light speed as `c`. An observer on Earth does *not* see the light moving at `0.9c + c = 1.9c`. They *also* measure the speed of that *same* light beam as `c`.
Consequence: To make these two postulates co-exist, the "common sense" ideas of absolute space and, most importantly, absolute time must be abandoned. Time and space are relative and interwoven.
These are the new equations that replace the classical Galilean transformations. They are derived from the two postulates and are the mathematical core of special relativity.
Consider two inertial frames: a "stationary" frame S (x, y, z, t) and a frame S' (x', y', z', t') moving with a constant velocity `v` along the +x axis relative to S.
The Lorentz Transformations are:
x' = γ (x - vt)
y' = y
z' = z
t' = γ (t - vx/c²)
Where γ (gamma) is the Lorentz factor:
γ = 1 / √(1 - v²/c²)
The "mixing" of space and time is clear in the equation for `t'`. It shows that time measured in one frame (`t'`) depends on both the time (`t`) and the *position* (`x`) in the other frame. This leads to the "Relativity of Simultaneity" - two events at different locations that are simultaneous in S (Δt=0) will *not* be simultaneous in S' (Δt' ≠ 0).
A direct consequence of the Lorentz transformations. It states that a moving object is measured to be shorter in its direction of motion.
Let L₀ be the "proper length" of an object (its length as measured in its *own* rest frame, e.g., by an astronaut holding a meter stick).
Let L be the length of that *same* object as measured by an observer who sees it moving at speed `v`.
The relationship, derived from the Lorentz transformations, is:
L = L₀ / γ = L₀ √(1 - v²/c²)
Since γ ≥ 1, the measured length L is always less than or equal to L₀.
The other major consequence. It states that a moving clock is measured to run slower than a stationary clock.
Let Δt₀ be the "proper time" interval (the time between two events that happen at the *same location* in a single frame, e.g., one tick of a clock at rest).
Let Δt be the time interval between those *same* two events as measured by an observer who sees the clock moving at speed `v`.
The relationship, derived from the Lorentz transformations, is:
Δt = γ Δt₀ = Δt₀ / √(1 - v²/c²)
Since γ ≥ 1, the measured time interval Δt is always greater than or equal to Δt₀.
In words: The stationary observer sees the moving clock taking *more* time (Δt) to complete one tick than the proper time (Δt₀). Therefore, the moving clock appears to run slow.