• The Centre of Mass of a Scalene Triangle

    Equilateral and isosceles triangles of constant mass per unit area have symmetry, and this symmetry can be used to find the centre of mass of a triangle. The centre of mass of an equilateral triangle is quickly found to be two thirds of the distance...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3655-the-centre-of-mass-of-a-scalene-triangle.html

  • The Centre of Mass of a Scalene Triangle

    Equilateral and isosceles triangles have symmetry, and this symmetry can be used to find the centre of the triangle. The centre of an equilateral triangle is quickly found to be two thirds of the distance from the vertex to the centre of the the...

    https://astarmathsandphysics.com/ib-maths-notes/vectors-lines-and-planes/1174-the-centre-of-mass-of-a-scalene-triangle.html

  • Centre of Mass of a Solid of Revolution

    When a curve is rotated about the - axis, the centre of mass of the solid generated will line on the – axis because of the symmetry of the curve. The centre of mass will not lie on the – axis however. If the centre of gravity of a volume of revolution...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3632-centre-of-mass-of-a-solid-of-revolution.html

  • Centres of Gravity

    Every body has a point through which the whole mass of the body can be considered to act. If the body is held tat the centre of gravity it will not move or tilt. If the body is suspended from some arbitrary point, the centre of gravity will hang below...

    https://astarmathsandphysics.com/o-level-physics-notes/143-centres-of-gravity.html

  • Centres of Gravity

    up of lots of individual atoms, the weight of it can be considered to act through a single point. This point is called the centre of gravity. Given any body, the position of the centre of gravity can be found by hanging it from several different points....

    https://astarmathsandphysics.com/ib-physics-notes/mechanics/1353-centres-of-gravity.html

  • Centre of Triangle Using Vectors

    can prove that the centre of a triangle is two thirds of the way from a vertex to the midpoint of the opposite side with vectors. Consider the triangle below, with \[\mathbf{AB}=\mathbf{b}, \; \mathbf{AC}= \mathbf{c}\] and \[P, \; Q\] as the midpoints...

    https://astarmathsandphysics.com/a-level-maths-notes/c1/5442-centre-of-triangle-using-vectors.html

  • Proof That a Line Drawn From the Centre of a Circle to a Chord Bisects the Chord

    The theorem is illustrated below. is the midpoint of The proof is quite simple. Draw radii to complete the triangles and These triangles have a side in common ( ) and Then from Pythagoras Theorem for triangle and for triangle

    https://astarmathsandphysics.com/gcse-maths-notes/629-proof-that-a-line-drawn-from-the-centre-of-a-circle-to-a-chord-bisects-the-chord.html

  • Proof That a Line Drawn From a Point to the Centre of a Circle Bisects the Angle Between the Two Tangents Drawn From That Point

    The theorem is illustrated below. Proof: Construct the triangles and by drawing radii as below. since both are radii of the circle and is common to both. Further, angle since these are between a tangent and a radius. From Pythagoras theorem, so and the...

    https://astarmathsandphysics.com/gcse-maths-notes/630-proof-that-a-line-drawn-from-a-point-to-the-centre-of-a-circle-bisects-the-angle-between-the-two-tangents-drawn-from-that-point.html

  • Proof That Every Point of an Open Ball in a Metric Space is the Centre of an Open Ball

    Theorem Let be an open ball in the metric space For every point there exists an open ball such that hence Define We must show Suppose then or equivalently hence is a metric so Hence

    https://astarmathsandphysics.com/university-maths-notes/topology/2283-proof-that-every-point-of-an-open-ball-in-a-metric-space-is-the-centre-of-an-open-ball.html

  • Centres of Mass, Integral Method

    We apply the formulae, To find the distances of the centre of gravity about certain axes. Suppose we want to find the distance of the centre of gravity of a uniform circular cone from it's base. By symmetry, and will be on a vertical line through the...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3633-centres-of-mass-integral-method.html

  • Centres of Mass, Summation Method

    For a uniform regular body with symmetry eg a cube, flat triangle, sphere, the centre of mass is in the middle. Unfortunately, most bodies are not regular or uniform. To find the centre of mass of an arbitrary body we must use one of the formulae, or...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3634-centres-of-mass-summation-method.html

  • Centres of Mass Tables

    Body Centre of Mass Solid hemisphere, radius from centre Hemispherical shell, radius from centre Circular arc of length radius from centre Sector of circle of arc length from centre Solid right circular cone, height from vertex Conical shell, height...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3635-centres-of-mass-tables.html

  • Orthogonal Trajectroes of System of Circles Centred on x Axis Passing Through the Origin

    the set of circles passing through the origin, with centres on the \[x\] axis. The equations have equations of the form \[(x-a)^2+y^2=a^2\] where \[a\] is the radius. Differentiating and simplifying gives \[x-a+y \frac{dy}{dx}=0 \rightarrow a=x+ y...

    https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5412-orthogonal-trajectroes-of-system-of-circles-centred-on-x-axis-passing-through-the-origin.html

  • Family of Curves Intersecting Circle Centre the Origin at 45 Degrees

    is the set of curves that intersect the set of circles centred at the origin with equations \[x^2+y^2=c\] at an angle of \[\pi /4\] ? Tangents to the circle at points \[(x,y)\] have gradient function \[\frac{dy}{dx}=- \frac{x}{y} tan \theta \] , where...

    https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5418-family-of-curves-intersecting-circle-centre-the-origin-at-45-degrees.html

  • Motion in a Circle

    as the the particle moves around the circle it must be making constant changes to it's velocity, always towards the centre. This means that it is always accelerating towards the centre, and implies from Newton's Second Law, F=ma, that it is accelerating...

    https://astarmathsandphysics.com/gcse-physics-notes/779-motion-in-a-circle.html

  • Definition of Terms Used For Lenses

    technical terms occur in discussions of lenses. The curvature of each surface of a lens makes it part of a sphere. The centre of curvature of the lens is the centre of this sphere. The principle axis is the line through the centre of the lens,...

    https://astarmathsandphysics.com/ib-physics-notes/optics-and-light/1391-definition-of-terms-used-for-lenses.html

  • Drawing Ray Diagrams For Thin Lenses

    axis. These will lie on the lens axis at a common distance form the lens. The distance from the focal points to the lens centre is labelled f , and is defined as positive for convergent lenses. There are 3 principal rays that are easy to track in order...

    https://astarmathsandphysics.com/igcse-physics-notes/385-drawing-ray-diagrams-for-thin-lenses.html

  • Constructions - Bisecting Angles and Lines - Constructing an Angle of 60 Degrees

    Bisecting an Angle To bisect the angle ABC draw arcs of equal length centred at B. Draw arcs of equal length centred at E and F to cross at G. The line BG bisects the angle. Constructing the Perpendicular Bisector to a Line To bisect the above draw...

    https://astarmathsandphysics.com/o-level-maths-notes/321-constructions-bisecting-angles-and-lines-constructing-an-angle-of-60-degrees.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 the centre, so the density of material at the centre of the Earth is far higher than the density of material at the surface. In fact,...

    https://astarmathsandphysics.com/igcse-physics-notes/430-the-density-of-the-earth.html

  • Uniform Circular Motion

    were not constantly changing, the object would move in a line. This means that the object is subject to a force towards the centre of the circle, O above. The force is supplied by the component of tension, towards the centre of the circle as shown in...

    https://astarmathsandphysics.com/ib-physics-notes/mechanics/1380-uniform-circular-motion.html

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