Problems

A particle of mass \(m\) is suspended by an elastic string of natural length \(l\), and is in equilibrium at a depth \(2l\) below the point of suspension. The particle is set in motion downwards with velocity \(\frac{an^2}{n-p}\), where \(n^2=\frac{g}{l}\), and simultaneously the point of suspension begins to oscillate with a motion given by \(z=a\sin pt\), \(z\) being measured vertically downwards. Prove that at time \(t\) the depth of the particle below its initial position is \[ \frac{an^2}{n^2-p^2}(\sin nt + \sin pt). \]

A particle moves in a plane under a central force \(\frac{\mu}{r^2}\) towards a point \(O\). Prove that the orbit is a conic. \par Find the equation of this conic if the particle is projected with velocity \(v\) from a point \(P\) at a distance \(a\) from \(O\), if the initial direction of the particle makes an angle \(\alpha\) with \(OP\) produced. Find also the velocity of the particle at each end of the major axis.

A uniform solid circular disc rests, with its plane vertical, on a planar lamina whose angle with the horizontal is \(\alpha\). The disc is released, and simultaneously the lamina begins to move downwards (without rotation) in its own plane, with uniform acceleration \(f\). If the disc rolls without slipping, shew that the velocity \(v\) of the centre of the disc, when it has moved through a distance \(l\), is given by \[ v^2 = \frac{2l}{3}(f+2g\sin\alpha). \]

Prove that the line joining a point \(P\) on the circumcircle of a triangle to the orthocentre of the triangle is bisected by the pedal line (or Simson's line) of \(P\) with respect to the triangle. Hence or otherwise shew that if \(ABCD\) is a quadrilateral, and \(AB, CD\) meet at \(P\), and \(BC, AD\) at \(Q\), then the orthocentres of the triangles \(QAB, QCD, PBC, PAD\) are collinear.

Three forces \(P, Q, R\) act along the sides \(BC, CA, AB\) of a triangle \(ABC\), and are in equilibrium with three forces \(P', Q', R'\) acting along \(HA, HB, HC\), where \(H\) is the orthocentre of the triangle. Prove that \begin{gather*} P\sec A + Q\sec B + R\sec C = 0, \\ PP'\operatorname{cosec} 2A + QQ'\operatorname{cosec} 2B + RR'\operatorname{cosec} 2C = 0, \\ \text{and } PP' + QQ' + RR' - (QR' + Q'R)\cos A - (RP' + R'P)\cos B - (PQ' + P'Q)\cos C = 0. \end{gather*}

Prove that if \(ABCD\) is a quadrilateral then in general the sum of the rectangles \(AB.CD\) and \(BC.AD\) is greater than the rectangle \(AC.BD\). State and prove under what condition the two quantities become equal. \(P\) and \(Q\) are two points within a quadrilateral \(ABCD\) such that the triangles \(BPC\), \(CQD\), \(BAD\) are similar, corresponding vertices being in the order named. Prove that \(APCQ\) is a parallelogram.

Shew that, if \[ (b-c)^2(x-a)^2 + (c-a)^2(x-b)^2 + (a-b)^2(x-c)^2 = 0, \] and no two of \(a, b, c\) are equal, then \[ x = \tfrac{1}{3} \{a+b+c \pm (a^2+b^2+c^2-bc-ca-ab)^{\frac{1}{2}}\}. \] Shew that one root of the equation \(x^3 = 100(x-1)\) is approximately 1.0103, and determine the other roots, correct to two places of decimals.

\(ABCDEFG\) is a regular heptagon inscribed in a circle of radius 1. Shew that the distance between the centroids of the triangles \(ABD, CEF\) is \(\frac{1}{3}\sqrt{7}\).

Nine equal light rods are smoothly jointed together at their ends so that three form a triangle \(BCD\), three others join the points \(B, C, D\) to a common point \(A\), and the remaining three join the points \(B, C, D\) to another common point \(E\). The framework is suspended from \(A\) and a load \(W\) hangs from \(E\). Find the stresses in all the rods.

Find the equation of the line joining the two points \(P\) and \(Q\) in which the circles \begin{align*} (x-a)^2 + y^2 &= a^2 \\ x^2 + (y-b)^2 &= b^2 \end{align*} intersect. Shew that the circle described on \(PQ\) as diameter is \[ (a^2+b^2)(x^2+y^2) = 2ab(bx+ay). \] Find the length of the tangent from the point \((\lambda b, \lambda a)\) to any one of these circles.