Knowledge Base

Hydromechanics

Black sea

Floating on salty water is possible due to its higher density, so the buoyancy force increases.
$ F_{A} = \rho_{M} \cdot V_{K in M} \cdot g $

Tree's height

Under ideal conditions (perfect vacuum) the maximum height water can rise in a tube is approx. 10 m
$ p_{W} = p_{0} \rightarrow \rho_{W} \cdot g \cdot h = 101.300 Pa $
This also would affect the height water can rise in trees and thus limit trees' maximum height. This limit is extended by suction forces caused by evaporation of water in the leafs to roughly 130m . To stabilize the effect no air may be inside the vascular bundles (see formula).

Rotational mechanics

Twirling water

The shape of water twirling inside a tank follows the equipotential surfaces of the combined gravity field and centrifugal force. The shape's height dependent on the distance from center of rotation is:
$ h(r) = h(0) + \frac{(\omega r)^2}{2g} $

Angular momentum

A body rotates faster when placing its point masses near its center of rotation because according to
$ \vec{L} = J \cdot \vec{\omega} $ and
$ J = \sum ( r^{2} \cdot m ) $
L must be conserved in quantity ( $ L=const. \rightarrow $ conservation of angular momentum), $ J $ decreases with smaller $ r $ and thus $ \omega $ must increase.

Forces depending on reference frame
  • Centripetal force:
    true force; directed towards center of rotation contrary to radial velocity; observed from inertial system; e.g. stiction force
    $ F_{CP} = -m \vec{\omega} \times (\vec{\omega} \times \vec{v_r}) $
  • Coriolis force:
    apparent force; contrary to tangential velocity; observed from rotating system when moving relatively to it
    $ F_{C} = -2m \vec{\omega} \times \vec{v_t} $
  • Euler force:
    apparent force; perpendicular to tangential velocity; observed from rotating system when axis of rotation or rotation speed are time variant
    $ F_{E} = -m \dot{\vec{\omega}} \times \vec{r} $
  • Optics

    Fermat's principle

    Light and particles (as waves) always propagate on paths between two points A and B which can be traversed in the least time. Light tries out every single possible path at the same time. That is due to the wave functions of all secondary waves travelling every possible path, while interfering among themselves. Only rays, including nearby ones, taking the route minimizing their travelling time interfere constructive with each other, the rest cancels out.
    So when looking into water for example the distorted geometry is due to light rays taking the shortest path available.

    Coherence

    If light is temporally coherent it is spectrally pure. Then the variance of the absolute value of the k-vectors is small.
    If light is spatially coherent it is not diffuse (direct radiation). Then the variance of the direction of the k-vectors is small.

    Celestial mechanics & Relativistics

    Free fall

    Earth's curvature is such that approx. every 8 km in horizontal direction it drops 5 m in vertical direction. Firing a bullet horizontally with 8 km/s it will fall 5 m downwards after one second, neglecting friction effects - it will never hit earh's surface again.
    In space astronauts always remain in a falling state around the Earth. Earth's gravity field acts as a source of momentum that flows into bodies' whole volume. The momentum current flowing into them can't flow out because they're not touching any body. Without the momentum current flowing out of their bodies they feel weightless.
    In order to take off from ground an airplane, bird, bee etc. has to give off at least the momentum gained at every moment through the gravity field to the surroundig air, back to Earth.

    Hubble's constant

    Moon increases its distance to Earth every year by 3 cm because of the expansion of the universe given by Hubble's constant
    $ H_{0} \approx 2.28475 \cdot 10^{-18} \dfrac{\frac{m}{s}}{m} $.

    Spacetime
  • The scalar curvature of space time at the event horizon of the Schwarzschild metric features an ellipsoid shape around the central singularity regarding rotating black holes.
  • A body not moving at all, propagates through time only. The faster its movement, the more it propagates through space. A photon always propagates through space only. Its proper time is standing still, therefore it's not "aging".
  • Time dilation

    At 86,60254038 % of vacuum light speed time passes two times slower than in a related rest frame.

    Speed of light

    The speed of light in vacuum is not always the highest possible one (299.792.458 m/s)! When light passes close to massive objects, its path gets bended. Thus it needs more time to travel along that path and so its speed decreases even in vacuum. Einstein had a big problem arguing why that happens. This caused him to develop his theory of general relativty.

    Quantum mechanics

    Quantum number

    The selection rule $\Delta l \pm 1 $ says that due to conservation of momentum the angular momentum of an atom has to change by ℏ [Nms] when a photon is involved in a transition (e.g. absorption, emission).

    Atom's empty space

    The space between electrons and the nucleus in an atom is not empty. Due to the wavelike behaviour of the electrons oscillating around the nucleus they aren't locally bounded. Thus only the matter density in this space is simply smaller.

    Thermodynamics

    Heat

    The phenomenon of heat conduction is based on "thermal sound", i.e. phonons - quantized oscillations of atoms in a solid's lattice.

    Temperature
  • Human skin can't feel temperatures at all. What it can feel is heat flux depending on the actual transfer mechanism. That's why it feels colder when walking with bare feet on stone or ceramic tiles, where the heat transfer is higher. Moreover you feel even colder when getting from warm inside to cold outside in winter, because the (negative) temperature gradient is higher in magnitude and so is heat flux.
  • Temperature is defined by the average kinetic energy of a particle given by
    $ E_{kin} = \frac{3}{2}k_{B}T $. So temperature is only an abstract expression for proportionality purposes (i.e. coefficient).
  • Thermal Radiation

    This type of radiation is only due to a body's temperature at which it emits EM-waves with respect to Planck's law of radiation.

    Electricity & Magnetism

    Electric current

    The electric current direction is is not the same as the direction of the flux $ \vec{j} $ but it is the vector of the average directional movement of the charge carriers $ \vec{v} $ (drift velocity vector) related by
    $ \vec{j} = \rho \vec{v} $.
    So the direction of $ \vec{j} $ remains the same no matter what charge the carrier (+ or -) has:
    $ \rho > 0 \land \vec{v} > 0 \longrightarrow \vec{j} > 0 $
    $ \rho < 0 \land \vec{v} < 0 \longrightarrow \vec{j} > 0 $
    Electric charge only flows from high to low potential, while charge carriers move in the direction dependent on their sign.

    Magnet behaviour

    Ideal soft-magnetic materials won't let a magnetic field permeate them ( $\vec{H}=0$ ). So do electric conductors with electric fields ( $\vec{E}=0$). An electro magnet works because the soft magnetic core expels the magnetic field from the inside. Ideal hard-magnetic materials however show up a constant magnetization ( $\vec{M}=const.$).

    Moving Charge Carriers in the Conductor (relativistic)

    Diverse

    Field theory

    The electromagnetic field is a physical system to which energy, momentum, angular momentum, entropy etc. can be assigned. Thus an EM-Wave isn't that "pure energy" that it's considered to be regarding e.g. particle/anti-particle annihilations.

    Resonance

    The meaning of resonance is that a system getting excited absorbs and dissipates maximum energy.

    Persue for energy minimum

    If a system with an initial energy content is at its energy minimum after a while (e.g. a pendulum), the production of entropy is zero and thus it can't dissipate energy anymore. So its initial amount of energy shrank through production of entropy accompanied by energy dissipation $ T \dot{S} $.

    Particle interactions at LHC (CERN)

    After the collision of protons in the particle accelerator, the new particles don't emerge from protons' fragments but are beeing reformed to particles directly out of the energy transported by the fast protons and condensed in a small spatial spot.

    Redefinitions

    Voltage \longrightarrow U = \frac{\dot{E}}{\dot{Q}}
    Velocity \longrightarrow v = \frac{\dot{E}}{\dot{p}}
    Temperature \longrightarrow T = \frac{\dot{E}}{\dot{S}}