The diagram represents a system of seven light rods smoothly jointed at \(A, B, C, D, E,\) and supporting a load \(W\) at \(C\). \(A\) is fixed and \(D\) rests against a smooth support, the horizontal and vertical rods being all of equal length. [The diagram shows a structure. A is a point. AB is horizontal to the right. BC is horizontal to the right, continuing AB. AE is vertical downwards. ED is horizontal to the right. B is connected to E. C is connected to E. A load W hangs from C. The lengths AB, BC, ED, AE are equal. D is against a vertical wall.] Construct a force-diagram shewing the reaction in each rod, and state in each case whether it is a tension or a thrust. Find also the pressure at \(D\), and the resultant action at \(A\).
A hollow triangular prism with open ends is formed from three rectangular sheets of metal of uniform thickness. Its cross section is a triangle \(ABC\), obtuse-angled at \(C\). Prove that, if placed with the face \(BC\) in contact with a horizontal plane, it will topple over if \[ 2a^2 < (c-b)(b+c-a). \]
Two beads \(A, B\), whose weights are \(w_1, w_2\) are tied to the ends of a string, on which is threaded a third bead \(C\) of weight \(W\). The beads \(A, B\) can slide on a rough horizontal rod, whose coefficient of friction with each bead is \(\mu\). If, when \(A, B\) are as far apart as possible, the strings \(AC, BC\) each make an angle \(\theta\) with the vertical, prove that, if \(w_1>w_2\), \[ \tan\theta = \mu \{1+2(w_2/W)\}. \]
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 \[ (5x^2 - 16x - a)^2 + b(x-1)^2 + c(x^2+x)^2 + d(x^2+2)^2 = 0. \]