Three rods \(OA, OB, OC\), each of length \(l\) and of equal weight, are smoothly jointed together at \(O\) and are placed symmetrically over a smooth sphere of radius \(a\), the joint \(O\) being vertically above the centre of the sphere, and the rods resting against its surface. Prove that, if \(\sqrt{2} \cdot l=3a\), the rods, when in equilibrium, will be mutually at right angles to one another.
A train whose mass is 200 tons starts from rest on a level track. Until the velocity reaches 12 miles an hour the engine exerts a constant pull equal to the weight of 5 tons, and throughout the motion the train is subject to a frictional resistance equal to the weight of 1 ton. In what time will the velocity of 12 miles an hour be attained, and at what horse-power will the engine be working at that instant? If, after attaining the velocity of 12 miles an hour, the engine continues to work at a constant horse-power, prove that the velocity of the train will gradually approach, but can never exceed, 60 miles an hour.
The height above the ground of a shot fired vertically upwards is given by the following table:
A particle is projected from a point at a distance \(a\) from a vertical wall, so that after striking the wall it rebounds and strikes the horizontal plane through the point of projection at a distance \(2a\) from the wall. Prove that, if the wall had not existed, the range of the projectile on a horizontal plane would have been equal to \((2+e)a/e\), the whole motion taking place in a plane perpendicular to the wall, and \(e\) being the coefficient of restitution between the particle and the wall.
Two equal particles \(A, B\) are tied to the ends of a string 9 feet long, which passes over a small pulley \(P\), fixed at a height of 4 feet above a smooth horizontal plane. Initially \(A\) is held at rest on the plane, \(AP=8\) ft., and \(PB\) is vertical and \(=1\) ft. If the system is now released, prove that \(B\) will strike the plane with a velocity \(\sqrt{(27g/17)}\). Shew also that the tension of the string at the beginning of the motion is equal to \(\frac{8}{17}\) of the weight of either particle.
Two particles \(A, B\), whose masses are \(m_1, m_2\), are tied to the ends of an elastic string whose natural length is \(a\), and they are placed on a smooth table so that \(AB=a\). If \(B\) is now projected with velocity \(v\) in the direction \(AB\), prove that the string will become slack after a time \[ \pi \sqrt{\frac{m_1 m_2 a}{(m_1+m_2)\lambda}}, \] and that the maximum value of the tension of the string is equal to \[ v\sqrt{\frac{m_1 m_2 \lambda}{(m_1+m_2)a}}, \] \(\lambda\) being the modulus of elasticity of the string.
Shew that, for certain integral values of the constants, the expression \[ (5x^2 - 16x - a)^2 + b(x-1)^2 + c(x^2+x)^2 + d(x^2+2)^2 = 0. \]
Solve the equations:
Expand in ascending powers of \(x\) the fraction \[ \frac{2x + (9+3x^2)^{1/2}}{3-x} \] as far as the fifth power of \(x\), and shew that, for small values of \(x\) it leads to a good approximation for \(e^x\). Deduce that \(e^{1/4} = 1.2840\dots\).
Prove that, in any triangle, \(a \cot A = b \operatorname{cosec} C - a \cot C\). If \(a=19.1, b=15.1, \tan C = 3\), find \(\tan A\) and \(\tan B\).