A straight uniform rod of length \(2l\) rests in contact with a small smooth fixed peg, the lower end of the rod being on a smooth plane which is inclined at an angle \(\alpha\) to the horizontal. If \(c\) is the shortest distance of the peg from the plane, and \(\theta\) the angle the rod makes with the vertical, prove that \[ l \sin\theta \cdot \cos^2(\theta-\alpha) = c\sin\alpha. \]
Three uniform freely jointed rods form an isosceles triangle \(ABC\). \(P\) is the weight of each of the equal rods \(AB, AC\), and \(Q\) is the weight of \(BC\). The figure is suspended from \(A\). Prove that the reaction between the rods at either of the lower hinges makes with the vertical an angle whose tangent is \(\displaystyle\left(1+\frac{Q}{2P}\right)\tan\frac{A}{2}\).
Prove that the greatest inclination to the horizontal at which a uniform rod can rest inside a rough sphere is given by \(\tan\theta = \displaystyle\frac{\sin\lambda.\cos\lambda}{\cos^2\alpha - \sin^2\lambda}\), where \(\lambda\) is the angle of friction, and the rod subtends an angle \(2\alpha\) at the centre of the sphere.
A fine elastic string \(OAB\), whose modulus of elasticity is \(\lambda\) and unstretched length is \(a\), has one end fixed at \(O\), and passes over a small smooth fixed peg at \(A\), where \(OA=a\). A particle of mass \(m\) hangs in equilibrium at \(B\). Prove that if a horizontal impulse \(I\) is applied to the particle, it will move in a horizontal straight line with simple harmonic motion of amplitude \(I\left(\displaystyle\frac{a}{\lambda m}\right)^{\frac{1}{2}}\).
Four heavy particles lie in a straight line on a smooth horizontal plane. The first is projected along the straight line to strike the second, which in turn strikes the third, which in turn strikes the fourth. The first three particles are now at rest, and the coefficient of restitution at each impact was \(e\). Prove that the final energy is \(e^3\) times the energy just before the first impact.
A particle slides down the outside of a fixed smooth sphere of radius \(r\), starting from rest at a height \(\frac{1}{8}r\), measured vertically, above the centre. Prove that it leaves the sphere when at a height \(\frac{1}{4}r\) above the centre. Prove also that when the particle is at a horizontal distance \(r\sqrt{2}\) from the centre, it is at a vertical depth \(4r\) below the centre.
If the tractive force per ton of an electric train at speed \(v\) is \[ \frac{a(b-v)}{c+v} \text{ tons weight}, \] where \(a, b\) and \(c\) are constants, find the speed \(V\) at which the horse-power exerted is a maximum. Find also the gradient up which, if friction and wind resistance are neglected, the maximum speed attainable is \(V\). \item[(i)] If \[ x^m y^n = (x+y)^{m+n}, \] prove that \[ \frac{dy}{dx} = \frac{y}{x}. \] \item[(ii)] Sketch roughly the shape of the curve \[ y^2 = x(x-1)(2-x), \] and prove that part of it is an oval of breadth 1, and depth \(\sqrt{\frac{4}{27}}\). Note: The scanned document depth value is hard to read. It's likely `sqrt(4/27)` or `sqrt(64/27)`. Using the value that seems more correct.
If a tree trunk \(l\) feet long is a frustum of a cone, the radii of its ends being \(a\) and \(b\) feet (\(a>b\)); and if it is required to cut from it a beam of uniform square section; prove that the beam of greatest volume is \(\displaystyle\frac{al}{3(a-b)}\) feet long.
Prove that
Nine thin rods, freely jointed together, are arranged so as to form an equilateral triangle \(ABC\) together with the triangle \(DEF\) constituted by joining the middle points of the sides. This framework is symmetrically supported in a vertical plane by vertical strings attached to the joints at \(B\) and \(C\). Show that, if the weight of each of the three inner rods is \(W\) and the weight of each of the six outer rods is \(w\), then there is a tension in the rod \(EF\) of magnitude \[ \frac{2}{\sqrt{3}}(W+w). \]