• Equation if Motion For an Inviscid, Incompressible Fluid

    For an incompressible fluid, so If the body force per unit volume is applying Newton's Second Law to a cube of unit volume to obtain We can also write where is the pressure in the fluid. Equating these gives (1) Consider the motion of a particle in the...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1569-equation-if-motion-for-an-inviscid-incompressible-fluid.html

  • Kelvin's Theorem

    Kelvin's circulation theorem states: In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant. where is the circulation around a material contour which may vary with...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1571-kelvin-s-theorem.html

  • Newtonian Fluids

    Viscosity may be thought of as a fluids resistance to the shearing forces acting on the fluid. We may picture a fluid between two plates, one stationary and one moving parallel to the first. The relative motion of the two plates cause shearing forces...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1572-newtonian-fluids.html

  • The Navier - Stokes Equation

    The Navier – Stokes equation models the behaviour of a fluid element because of the forces acting on it, including viscous forces. When the viscous forces are ignored, the equations become Euler's equation. The equation is difficult to solve, and...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1577-the-navier-stokes-equation.html

  • Limit Cycles

    For first order systems the motion tends to fixed points or infinity, but for second order systems the situation can be more complex. Consider for example a system separable in polar coordinates such that the motion has a fixed point (in terms of )...

    https://astarmathsandphysics.com/university-physics-notes/non-linear-dynamics/1599-limit-cycles.html

  • The Boltzmann Distribution

    The distribution of the energies of a collection of particles depends on the average energy, which depends on the temperature. The most probable distribution of the energies of a collection of particles depends on the temperature therefore. The...

    https://astarmathsandphysics.com/university-physics-notes/physical-chemistry/1622-the-boltzmann-distribution.html

  • Probability Current or Probability Flux

    The probability density of a particle with wavefunction – or statefunction – is The probability density function changes in space, but it may also change in time. If the probability density is a function of time, then the particle will be moving and...

    https://astarmathsandphysics.com/university-physics-notes/quantum-mechanics/1635-probability-current-or-probability-flux.html

  • Bohr's Model of the Hydrogen Atom

    Bohr made the major innovation of hypothesising the quantization of angular momentum in units of where This means we can write We can use the ordinary rules of classical Newtonian mechanics to derive the equation giving the differences in the energy...

    https://astarmathsandphysics.com/university-physics-notes/quantum-mechanics/1646-bohr-s-model-of-the-hydrogen-atom.html

  • The Leonard - Jones Potential

    The Lennard-Jones or L-J potential is a mathematically simple model that describes the interaction between a pair of neutral atoms or molecules. The expression of the L-J potential is where is the depth of the potential well, is the (finite) distance...

    https://astarmathsandphysics.com/university-physics-notes/quantum-mechanics/1651-the-leonard-jones-potential.html

  • The Stern Gerlach Experiment

    An electron, being a particle with an associated magnetic moment, may be pictured as a little spinning top. In the presence of a magnetic field this magnetic moment will align with the field in one of two ways – either spin up or spin down. The...

    https://astarmathsandphysics.com/university-physics-notes/quantum-mechanics/1656-the-stern-gerlach-experiment.html

  • Compton Scattering of X - Rays

    We can consider photons to carry momentum. As in any collision involving particles with non zero mass, collisions involving photons and matter particles will result in the overall conservation of momentum and energy. A photon with wavelength can be...

    https://astarmathsandphysics.com/university-physics-notes/special-and-general-relativity/1659-compton-scattering-of-x-rays.html

  • Transformation of the Alpha Factor Between Inertial Frames

    Suppose we have two inertial frames O and O' with O' moving along the x axis of O with a constant speed A particle moving with a velocity in the inertial frame O has a momentum where The same particle in the inertial frame O' has a momentum where In...

    https://astarmathsandphysics.com/university-physics-notes/special-and-general-relativity/1673-transformation-of-the-alpha-factor-between-inertial-frames.html

  • Alternative Formulation of the Second Law of Thermodynamics

    The thermodynamic state of a substance is defined by the quantities – volume, – pressure, – Temperature and – internal energy. These four quanties are related by the equations and one equation for the internal energy eg This means we can choose any two...

    https://astarmathsandphysics.com/university-physics-notes/thermal-physics/1676-alternative-formulation-of-the-second-law-of-thermodynamics.html

  • Automorphisms

    An automorphism is an isomorphism from a group G onto itself. Example: If then is an automorphism of the group of complex numbers under addition. We test the requirements one by one. 1. With 2. If then and so is one to one. 3. is onto since if then and...

    https://astarmathsandphysics.com/university-maths-notes/abstract-algebra-and-group-theory/1684-automorphisms.html

  • The Lipschitz Condition

    Definition(Lipschitz): A continuous function where is called locally Lipschitz in if for all there is a neighbourhood of and a constant such that for all we have (and in general for metrics and in metric spaces and where is a continuous map from to )....

    https://astarmathsandphysics.com/university-maths-notes/advanced-calculus/1730-the-lipschitz-condition.html

  • The Method of Characterisitcs Explained

    To illustrate the method of characteristics consider the partial differential equation for Obviously, this is in fact an ODE. The solution is found by integration with respect to and reads where is an arbitrary function of is uniquely determined by...

    https://astarmathsandphysics.com/university-maths-notes/advanced-calculus/1731-the-method-of-characterisitcs-explained.html

  • Paramagnetism

    In an atom, most of the various orbital and spin magnetic moments of the electrons add up to zero. For some metals the atom has a net magnetic moment of the order of When the metal is placed in a magnetic field the field exerts a torque on each...

    https://astarmathsandphysics.com/university-physics-notes/electricity-and-magnetism/1551-paramagnetism.html

  • The Legendre Symbol

    Let be an odd prime and The Legendre symbol is defined as For example, and The Legendre symbol has the following properties for where is an odd prime. If then The Legendre symbol can be used to decide if some equations have solutions. If the equation...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/1979-the-legendre-symbol.html

  • Volume Traced Out by Moving Circle

    AA moving circle with variable size has plane parallel to the \[yz\] plane, passes through the line \[y=0, \; z=a\] and has a chord in common with the circle \[x^2+y^2=a^2, \; z=0\] , As the circle moves it generates a volume. To meet the conditions at...

    https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5384-volume-traced-out-by-moving-circle.html

  • Proof of Stefan Boltzmann Law

    According to Planck's radiation law, the radiation density is \[I= \frac{2 h}{c^2} \int^{\infty}_0 \frac{f^3}{e^{\frac{hf}{kT}} -1} df=\frac{2 h}{c^2} \int^{\infty}_0 \frac{f^3}{1-e^{- \frac{hf}{kT}}} e^{-\frac{hf}{kT}} df\] . Substitute \[\frac{hf}{kT}...

    https://astarmathsandphysics.com/university-physics-notes/quantum-mechanics/5401-proof-of-stefan-boltzmann-law.html

Results 161 - 180 of 209