A light string passes over a small smooth fixed pulley and to one end is attached a mass \(M\) and to the other a second small light pulley over which passes a second string carrying masses \(m_1\) and \(m_2\) at its ends. Find the condition that if the system is released from rest the mass \(M\) will not move, and determine the total downward force on the fixed pulley.
Explain what is meant by simple harmonic motion. Derive and solve the differential equation of such motion. A particle is describing simple harmonic motion with period \(2\pi/n\) and its velocities at two points distance \(h\) apart are \(u\) and \(v\). Show that the square of the amplitude of the motion is \[ \tfrac{1}{4}\{h^2 + 2(u^2+v^2)/n^2 + (u^2-v^2)^2/n^4h^2\}. \]
A fire-engine working at a rate of \(E\) horse-power pumps \(w\) cubic feet of water per second from a part of a reservoir at depth \(d\) feet below the open end of the hose. If the hose is held at an angle \(\alpha\) to the horizon, find the maximum height that the resulting jet of water can reach. 1 H.P. = 550 ft.-lb. per sec., 1 cu. ft. of water weighs 62.5 lb.
Three particles each of mass \(m\) are situated instantaneously at the vertices \(A,B,C\) of a triangle and are moving in such a way that after time \(t\) they all collide simultaneously at a point \(P\) in the plane \(ABC\) and coalesce to form a single particle of mass \(3m\). Show that the kinetic energy lost is \(m(BC^2+CA^2+AB^2)/6t^2\).
Two equal heavy beads \(A, B\) each of mass \(m\) move on a smooth horizontal wire in the form of a circle of radius \(a\) and centre \(O\). They are joined by a light spring of natural length \(2a\sin\alpha\) and modulus of elasticity \(\lambda\). If the angle \(AOB\) is denoted by \(2\theta\) show that during the motion \[ ma\sin\alpha\,\dot{\theta}^2 + \lambda(\sin\theta-\sin\alpha)^2 \] remains constant. If when the spring is at its greatest compression \(\theta=\beta\), show that maximum extension occurs when \(\sin\theta = 2\sin\alpha - \sin\beta\). What happens if \(2\sin\alpha-\sin\beta>1\)?
Prove that the sum of the roots of the equation \[ \begin{vmatrix} x & h & g \\ h & x & f \\ g & f & x \end{vmatrix} = 0 \] is zero, and that the sum of the squares of the roots is \[ 2 (f^2+g^2+h^2). \] Taking \(f, g, h\) to be real, and assuming that the roots are then all real, prove that no root exceeds \[ 2\sqrt{\tfrac{1}{3}(f^2+g^2+h^2)} \] in absolute value. In what circumstances (if any) can a root be equal to this in absolute value?
If \[ f(\theta) = \sum_{r=1}^n a_r \sin (2r-1)\theta, \] where \(a_1 > a_2 > \dots > a_n > 0\), show by considering \(f(\theta)\sin\theta\), or otherwise, that \[ f(\theta) > 0 \quad (0 < \theta < \pi). \]
A conic \(S\) and three points \(A, B, C\) are given in a plane. A variable point \(P\) is taken on \(S\), the line \(PC\) meets \(S\) again in \(Q\), the line \(QA\) meets \(S\) again in \(R\), and the line \(RB\) meets \(S\) again in \(P'\). By considering the relationship thus set up between \(P\) and \(P'\), or otherwise, prove that in general two triangles \(PQR\) (real or imaginary) can be inscribed in \(S\) so that \(QR, RP, PQ\) pass through \(A, B, C\), respectively. Give a construction for these two triangles, using only straight lines joining known points, and intersections of known lines with one another or with \(S\). Use as few lines as you can.
A conic \(U\) passes through two points \(X, Y\). Show that, by taking \(X, Y\) as two vertices of a triangle of reference \(XYZ\), we can in general write the equation of \(U\) in the form \[ xy=zu, \] where \(u\) is a homogeneous linear function of \(x, y, z\). Hence, or otherwise, prove that, if three (non-degenerate) conics have two points \(X, Y\) in common, the three common chords, not passing through \(X\) or \(Y\), of the three conics taken in pairs are concurrent.
The polynomial \(P(x)\) is defined, for a given positive integer \(n\), by \[ P(x) = \frac{d^n y}{dx^n}, \] where \(y=(x^2-1)^n\). Find the values of \(P(0)\), \(P(1)\), \(P(-1)\). Prove that \[ (x^2-1)P''(x) + 2xP'(x) - n(n+1)P(x) = 0. \]