• The Jeans Mass

    The study of collapsing gas clouds in space is important because they may collapse to form stars. A gas cloud has a self gravity which tends to cause the cloud to contract.This is opposed by gas pressure inside the cloud which will tend to make the...

    https://astarmathsandphysics.com/university-physics-notes/astronomy/1512-the-jeans-mass.html

  • Preservation of Area by Time Independent Systems

    A system influenced by time dependent forces or which is represented in a rotating or non inertial reference frame has a Hamiltonian which depends explicitly on time, The rate of change of the Hamiltonian is given by On using Hamilton's equations of...

    https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1529-preservation-of-area-by-time-independent-systems.html

  • Principle of Least Action

    The principle of least action is a variational principle that can be used to obtain the equations of motion for that system in a way that is independent of Newtonian mechanics. The action principle is preceded by earlier ideas in surveying and optics....

    https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1530-principle-of-least-action.html

  • Time Dependent Canonical Transformations

    Time dependent canonical transformations are very similar to the time independent case, and as with the time independent case, the preservation of area is a part of the analysis. Consider the transformation The condition for such a transformation to...

    https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1541-time-dependent-canonical-transformations.html

  • Using Ampere's Law to Find the Magnetic Field Around a Long Straight Current Carrying Wire

    Ampere's Law equates the line integral of the magnetic field around an arbitrary loop to %mu-0 *the current passing through the loop: The loop may be any closed loop. If no current passes through the loop or if it passes through and back without...

    https://astarmathsandphysics.com/university-physics-notes/electricity-and-magnetism/1555-using-ampere-s-law-to-find-the-magnetic-field-around-a-log-straight-current-carrying-wire.html

  • Error Anaysis 2

    Error has to do with uncertainty in measurements that nothing canbe done about. A metre rule might measure to the nearest mm. If ameasurement is repeated, the length obtained may differ and none ofthe measurements can be preferred. Although it is not...

    https://astarmathsandphysics.com/university-physics-notes/experimental-physics/1560-error-analysis-2.html

  • Classificatin of Open Channel Flow

    Consider the steady, uniform flow of an inviscid, incompressible liquid in an open channel with a rectangular cross section of constant width. If the width is and the depth is the volume flow rate where is the speed of the liquid. is constant from the...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1564-classificatin-of-open-channel-flow.html

  • Paths of Particles in Water Waves

    The velocity potential for water waves in water of finite depth, h, can be written From this we obtain and Integrating these equations with respect to gives and then This is the equation of an ellipse. For large and close to 0, far from the bottom...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1574-paths-of-particles-in-water-waves.html

  • Phase Curves and Phase Diagrams

    The motion of a dynamical system traces out a curve in phase space called a phase curve. For an autonomous system there is only one phase curve through each point, given by – with and denoting the initial conditions. Another solution represents the...

    https://astarmathsandphysics.com/university-physics-notes/non-linear-dynamics/1600-phase-curves-and-phase-diagrams.html

  • Quantum Tunnelling

    The particle has total energy and is incident on the barrier from the left. The Schrodinger equation in each of regions I, II and III takes the form In region I so The solution is where The term represents motion to the right and the term represents...

    https://astarmathsandphysics.com/university-physics-notes/quantum-mechanics/1638-quantum-tunnelling.html

  • Parallax

    In six months the Earth moves around the Sun from 1 to 2. is small so we can use the approximation where 1 Astronomical Unit=150 million km, the distance from the Earth to the Sun, and is in radians. Example

    https://astarmathsandphysics.com/university-physics-notes/the-universe-and-cosmology/1680-parallax.html

  • d Alembert's Solution to the Wave Equation

    A particularly neat solution to the wave equation, valid when the string is so long it may be approximated by one of infinite length, was obtained by d’Alembert. The idea is to change coordinates from and to and in order to simplify the equation....

    https://astarmathsandphysics.com/university-maths-notes/advanced-calculus/1723-d-alembert-s-solution-to-the-wave-equation.html

  • Proof of Snell's Law

    When light passes from a medium to an optically denser medium the path of the light is refrated towards the normal. Suppose the speed in medium 1 is \[v_1\] and the speed in medium 2 is \[v_2\] then the time taken to pass from point A to B above is \[t=...

    https://astarmathsandphysics.com/university-physics-notes/light-and-optics/5381-proof-of-snell-s-law.html

  • Cartesian Equation of Motion From a Vector Differential Equation

    We can solve vector differential equations by writing separate differential equations for each component. Solve these differential equations, then add them with the appropriate coordinate vectors to the give the answer to the question. A Cartesian...

    https://astarmathsandphysics.com/a-level-maths-notes/m5/5074-cartesian-equation-of-motion-from-a-vector-differential-equation.html

  • Constructing Triangular Numbers From Square Numbers

    If a number of one of the forms \[2n^2-1, \; 2n^2+1\] is square ( \[{}=m^2\] )then \[(nm)^2\] is triangular, since \[(nm)^2=n^2m^2=n^2(2n^2-1)=\frac{2n^2}{2}(2n^2-1)=\frac{k}{2}(k-1)=\frac{j}{2}(j+1)\] where first \[k=2n^2\] then \[k=j+1\] . or...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5210-constructing-triangular-numbers-from-square-numbers.html

  • Parts of a Circle

    Labels for the different parts of a circle are given below Radius – the radius of a circle is the distance from the centre labelled O to the circumference. The radius is labelled The circumference of a circle is the perimeter, and is labelled Diameter...

    https://astarmathsandphysics.com/gcse-maths-notes/615-parts-of-a-circle.html

  • Solving Equatins Graphically

    Suppose we sketch the graph We obtain the graph below. We can use the graph to solve the equation by finding the intersection of the graph with the – axis, since this is where Suppose we want to solve a different equation – for example, We could sketch...

    https://astarmathsandphysics.com/gcse-maths-notes/657-solving-equatins-graphically.html

  • Solving Exponential Simultaneous Equations Algebraically

    If some quantity is increasing decreasing by the same proportion or factor in each time interval, then it is said to vary exponentially. A good example is money left in a bank which attracts compound interest at the same rate each year. If the rate of...

    https://astarmathsandphysics.com/gcse-maths-notes/659-solving-exponential-simultaneous-equations-algebraically.html

  • Accoustics

    The reverberation of sound is particularly noticeable in cathedrals and other large buildings where multiple reflections can occur rom the walls, roof and floor. In St Paul's Cathedral in London it can take about 6 seconds for the notes of the organ to...

    https://astarmathsandphysics.com/o-level-physics-notes/132-accoustics.html

  • Eclipses

    The moon orbits the Earth. The Earth - moon combination orbits the Sun. Sometimes the Sun, Earth and Moon are lined up, meaning that either the Earth or moon is at least partially in the shadow of the other. These instances are called eclipses. If the...

    https://astarmathsandphysics.com/o-level-physics-notes/154-eclipses.html

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