A particle of mass \(m\), free to move without friction in a circular tube of radius \(a\) in a vertical plane, is attached to the highest point of the tube by an elastic string of length \(l(<\pi a)\) and modulus \(\lambda\) lying in the tube. Shew that the system is in equilibrium when \(\theta\), the angular extension of the string, is the least positive root of the equation \[ a\lambda\theta - mgl\sin\left(\theta+\frac{l}{a}\right) = 0. \] If there are any other positive roots of this equation, what is their significance?
A particle of mass \(m\) is at rest on top of a smooth sphere of radius \(a\). The sphere is fixed on a smooth horizontal plane. The particle is slightly displaced; shew that it will leave the sphere when the radius to the particle makes an angle \(\theta\) with the upward drawn vertical given by the equation \(\cos\theta = \frac{2}{3}\). If the plane is perfectly elastic, shew that the particle rises to a height \(\frac{50}{27}a\) after each rebound.
A train of mass \(M\) is pulled by its engine against a constant resistance \(R\). The engine works at constant power, doing \(H\) units of work per second. Shew that the time taken to increase the velocity from \(v_0\) to \(v_1\) feet per second is equal to \[ \frac{MH}{R^2} \log\frac{H-Rv_0}{H-Rv_1} - \frac{M}{R}(v_1-v_0) \text{ seconds.} \]
A particle moves in a parabola, whose focus is \(S\), under the action of gravity. Prove that when the particle is at \(P\) the components of its velocity along and perpendicular to \(SP\) are respectively equal to the vertical and horizontal components of its velocity. Shew that the component of its acceleration in the direction \(SP\) is equal to \(g-u^2/r\), where \(r=SP\) and \(u\) is the velocity of the particle at the vertex of the parabola.
A particle of mass \(m\) is attached by an inelastic string of length \(l\) to the top of a high pole. It is thrown upwards into the air from a point at a distance \(l\) from the top of the pole and at a height \(d\) vertically above it. Its initial velocity is \(\sqrt{(gd)}\), and its initial direction is at right angles to the string. Shew that when the string becomes taut it exerts an impulse equal to \(\frac{1}{8}m\sqrt{(gd)}\frac{d h^3}{l^4}\) on the particle, where \(h=\sqrt{(l^2-d^2)}\).
A light bar \(OA\) of length \(2a\) with a particle of mass \(m\) attached to its middle point turns in a horizontal plane about a vertical axis through \(O\); and a light bar \(AB\), of the same length as \(OA\) and with a similar particle attached to its middle point, is freely jointed at \(A\) to the bar \(OA\). A smooth guide compels the end \(B\) to move along a horizontal straight line \(Ox\). The angle \(AOx=\theta\). Shew that \[ \frac{d^2\theta}{dt^2} + \frac{4\omega^2\sin\theta\cos\theta}{(5-4\cos^2\theta)^2} = 0, \] where \(\omega\) is the value of \(\frac{d\theta}{dt}\) when \(\theta=0\).
(i) If the remainders when a polynomial \(f(x)\) is divided by \((x-a)(x-b)\) and by \((x-a)(x-c)\) are the same, shew that \[ (b-c)f(a) + (c-a)f(b) + (a-b)f(c) = 0. \] (ii) When \(x\) and \(y\) are eliminated from the equations \[ x^2-y^2 = ax-by; \quad 4xy=bx+ay; \quad x^2+y^2=1, \] prove that \[ (a+b)^{\frac{2}{3}} + (a-b)^{\frac{2}{3}} = 2. \]
In the series \(u_0+u_1x+u_2x^2+\dots\) any three successive coefficients are connected by the relation \[ u_{r+1} + pu_r + qu_{r-1}=0; \] shew how to find the sum to \(n\) terms. Assuming that the series \[ 2+\frac{7}{5}x + \frac{91}{125}x^2 + \dots \] is of this type, find the \(n\)th term and the sum to infinity.
With the usual notation for the radii of the inscribed and escribed circles of the triangle \(ABC\), prove that
Prove that \[ \sin(\alpha+\beta)+\sin(\alpha+2\beta)+\dots+\sin(\alpha+n\beta) = \frac{\sin(\alpha+\frac{n+1}{2}\beta)\sin\frac{n\beta}{2}}{\sin\frac{\beta}{2}} \] and deduce the sum of \[ \sin\theta - \sin2\theta + \sin3\theta - \dots - \sin 2r\theta. \] Shew also that \[ \cos^2x + \cos^22x + \dots + \cos^2nx = \frac{2n-1}{4} + \frac{\sin(2n+1)x}{4\sin x}. \]