• Hamiltonian Systems

    are and The fixed points of this system are and The fixed points of a Hamiltonian system can only be a saddle or a centre, since the linearisation matrix is given by The eigenvalues are the solution to for this linearisation so we solve This has...

    https://astarmathsandphysics.com/university-physics-notes/non-linear-dynamics/1595-hamiltonian-systems.html
  • Bodies on Slopes Toppling

    A body will topple when the vertical through the centre of gravity lies just outside the base. A particle on a slope is typically more susceptible to toppling because the vertical is closer to the point where it would lie just outside the base. The...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3630-bodies-on-slopes-toppling.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
  • University Maths Notes - Number Theory

    https://astarmathsandphysics.com/university-maths-notes/number-theory.html
  • Proof that the Focal Length of a Spherical Mirror is Half the Radius of Curvature

    pass through on reflection. This is the distance in the diagram below. The radius of curvature of a spherical mirror is the centre of the sphere of which the mirror is a part. Any ray of light passing through the centre of curvature will strike the...

    https://astarmathsandphysics.com/o-level-physics-notes/206-proof-that-the-focal-length-of-a-spherical-mirror-is-half-the-radius-of-curvature.html
  • Stable, Unstable and Neutral Equilibrium

    Ito illustrate the difference between the kinds of equilibrium consider a cone. If the cone can be stood so that it's centre of gravity is vertically above the point of contact with the ground, (a) above, then theoretically the cone will be in...

    https://astarmathsandphysics.com/o-level-physics-notes/225-stable-unstable-and-neutral-equilibrium.html
  • The Density of the Earth

    pretty obvious that the densest materials sink. In fact,over the lifetime of the Earth, they have sunk all the way to thecentre, so the density of material at the centre of the Earth is farhigher than the density of material at the surface. In fact, as...

    https://astarmathsandphysics.com/gcse-physics-notes/842-the-density-of-the-earth.html
  • Radians and Degrees

    to a turn through radians or and half a turn is equivalent to a turn through (Remember that 360° or is the angle at the centre of a circle and 180° or is the angle on a straight line). We can write or We can change radians into degrees using We also...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1138-radians-and-degrees.html
  • Circular Vertical Motion

    direction of the force. At any time the particle is moving tangentially to the circle and the force is directed towards the centre of the circle – along it's radius. Tangent and radius being at right angles, the particle never moves in the direction of...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3638-circular-vertical-motion.html
  • Motion of Particle on Inner Surface of Smooth Sphere

    speed It will move along the internal surface of the sphere. Suppose at some point B it has a speed If the line from the centre of the circle to the point B makes an angle with the upwards vertical then the force of gravity will have a component towards...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3644-motion-of-particle-on-inner-surface-of-smooth-sphere.html
  • Motion of a Particle on Surface of Smooth Sphere

    at which the particle loses contact with the sphere. If at some point the line joining the position of the particle to the centre of the circle makes an angle %theta with the downward vertical then the component of gravity towards the centre of the...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3645-motion-of-a-particle-on-surface-of-smooth-sphere.html
  • Maximum Area of Rectangle Inscribed in Semicircle

    a rectangle that can be inscribed in a semicircle of radius \[r\] ? Let the triangle OBC subtend an angle \[\theta\] at the centre of the circle. The triangles ABO abd CDO have the same area by symmetry, and together subtend an angle of \[\pi - \theta\]...

    https://astarmathsandphysics.com/a-level-maths-notes/c2/5322-maximum-area-of-rectangle-inscribed-in-semicircle.html
  • Angle of Tangent to Circle With x Axis

    To find the angle between the \[x\] axis and the tangent to the circle in the diagram, draw a line from the origin to the centre of the circle, a line from the centre of the circle to the \[x\] axis, and a radius from the centre of the circle to the...

    https://astarmathsandphysics.com/a-level-maths-notes/c2/5366-angle-of-tangent-to-circle-with-x-axis.html
  • Constructing a Set of Points a Fixed Distance From a Given Line

    To construct the set of points a fixed distance x from a line AB, with a set of compasses draw circles of radius x with centres at A and B, then draw tangents from one circle to another, parallel to the line. For example, construct the set of points 2...

    https://astarmathsandphysics.com/igcse-maths-notes/462-constructing-a-set-of-points-a-fixed-distance-from-a-given-line.html
  • Constructing the Set of Points a Fixed Distance From a Given Line

    To construct the set of points a fixed distance x from a line AB, with a set of compasses draw circles of radius x with centres at A and B, then draw tangents from one circle to another, parallel to the line. For example, construct the set of points 2...

    https://astarmathsandphysics.com/gcse-maths-notes/576-constructing-the-set-of-points-a-fixed-distance-from-a-given-line.html
  • How Dielectrics Work

    electric field – as in the diagram below. This can be reduced to a simple dipole by finding the distance between the centres of positive and negative charge. The effect is that the microscopic electric field is reduced by where is the charge on an...

    https://astarmathsandphysics.com/university-physics-notes/electricity-and-magnetism/1547-how-dielectrics-work.html
  • Locus of Points Equidistant Between Two Points

    locus will be a straight line halfway between the two points. We can construct this line by drawing circles of equal radius centred at A and B. The radius should be over half the distance AB. These two circles will intersect at two points. If you draw a...

    https://astarmathsandphysics.com/igcse-maths-notes/483-locus-of-points-equidistant-between-two-points.html
  • Locus of Points Equidistant Between Two Points

    locus will be a straight line halfway between the two points. We can construct this line by drawing circles of equal radius centred at A and B. The radius should be over half the distance AB. These two circles will intersect at two points. If you draw a...

    https://astarmathsandphysics.com/gcse-maths-notes/605-locus-of-points-equidistant-between-two-points.html
  • Enlargements - Negative Scale Factors

    than that shape to be enlarged. The scale factor k is positive, with the enlarged shape being k times as far from the centre of enlargement as the original shape. If the scale factor is more than one, the enlarged shape is bigger. If the scale factor is...

    https://astarmathsandphysics.com/gcse-maths-notes/646-enlargements-negative-scale-factors.html
  • Summary - Transformations - Translations - Rotations - Reflections - Enlargements

    in the direction and in the direction and is written A rotation is defined by three things: clockwise or anticlockwise, the centre of rotation, written and the angle of rotation in degrees. A reflection is described by the line of reflection or the...

    https://astarmathsandphysics.com/gcse-maths-notes/679-summary-transformations-translations-rotations-reflections-enlargements.html

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