One end of a beam, of length \(2a\), rests against a smooth vertical wall, and the beam is in contact with a thin smooth horizontal rail parallel to the wall at a distance \(b\) from it; prove that there is a position of equilibrium in which the inclination of the beam to the wall is given by the equation \[ \sin^3\theta = b/a \] but that this position is unstable.
The distance between the axles of a railway truck is \(a\) feet, and the centre of gravity is halfway between them and at a perpendicular distance \(b\) feet from the rails. With the lower wheels locked it is found that the greatest incline upon which the truck can rest is \(\alpha\). Prove that the coefficient of sliding friction between the wheels and the rails is given by \[ \mu = \frac{2a\tan\alpha}{a+2b\tan\alpha}. \]
The weight of a suspension bridge is so arranged that the total load carried by the chains including their own weight is uniformly distributed across the span of the bridge. Shew that the chains hang in parabolas. The span of the bridge is 200 feet and the sag at the middle is 10 feet. If the total load on each chain is 50 tons, find the greatest tension in each chain.
The horse power required to propel a steamer of 10,000 tons displacement at a steady speed of 20 knots is 15,000. If the resistance is proportional to the square of the speed and the engines exert a constant propeller thrust at all speeds, find the acceleration when the speed is 15 knots.
At a point on the ground from which a gun is fired the elevation of the top of a tower is \(\theta\). The gun is fired at an elevation \(\alpha\) and the shot strikes the tower at a point whose elevation is \(1'\) less than \(\theta\). Prove that in order that the shot may strike the top the elevation must be increased by \[ \frac{\sec^2\theta}{1+2\tan\alpha\tan\theta-\tan^2\alpha} \text{ minutes.} \]
Explain what is meant by the principle of the conservation of energy. The ends of an elastic string of natural length \(a\) and modulus \(\lambda\) are fixed at two points on a smooth horizontal table at a distance \(a\) apart. A particle of mass \(m\) is attached to the middle point of the string and is struck by a blow \(P\) in a direction perpendicular to the string. Shew that the greatest extension of the string in the subsequent motion is \(P\sqrt{a/m\lambda}\), and find the velocity in any position.
Four equal rods without mass are freely jointed at their extremities so as to form a framework, at each angular point of which is fixed a particle of mass \(m\). The whole is held in the form of a square with one diagonal vertical and dropped on to a horizontal table. Assuming the system to be inelastic, shew that the velocity of the top particle is unaltered by the impact.
An inclined plane of mass \(M\) is capable of moving freely on a smooth horizontal plane. A perfectly rough sphere is placed on its inclined face and rolls down. If \(m\) is the mass of the sphere, \(x'\) the horizontal space advanced by the inclined plane, \(x\) the distance on the plane rolled over by the sphere, shew that \[ (M+m)x' = mx\cos\alpha, \quad \frac{7}{5}x-x'\cos\alpha = \frac{1}{2}gt^2\sin\alpha, \] where \(\alpha\) is the inclination of the plane to the horizontal.
Explain what is meant by a recurring series and define the scale of relation of such a series. How may the sum to \(n\) terms of the series be found? Sum to \(n\) terms the series:
If the equations \(x^3+px^2+qx+r=0\) and \(x^2+ax+b=0\) have a common root, prove that \[ \begin{vmatrix} 1 & 0 & 1 & a-p \\ a & 1 & p & b-q \\ b & a & q & -r \\ 0 & b & r & 0 \end{vmatrix} = 0. \] If the equations \(x^3+2x^2+3x+6=0\) and \(x^2+ax-6=0\) have a common root, find the possible values of \(a\) and the corresponding common roots.