Shew that any force in the plane of a triangle is equivalent to three forces along the sides of the triangle, and shew how these forces may be determined. If the given force acts through the c.g. of the triangle \(ABC\) perpendicular to \(BC\), shew that the three forces along the sides which are equivalent to it are proportional to \(2a(b^2-c^2)\), \(b(b^2-c^2-3a^2)\), \(c(b^2-c^2+3a^2)\).
Five light rods are freely jointed so as to form a rectangle \(ABCD\) with a diagonal \(AC\). The framework is supported with \(A\) vertically above \(D\) by a horizontal force at \(A\) and a force at \(D\), and given weights are suspended at \(B, C\). Draw a force diagram shewing the stresses in all the rods.
Equal particles of weight \(W\) are knotted to a string which is suspended from two fixed points in such a manner that all the horizontal distances between successive particles are equal to \(a\). Prove that the vertical distance between the first and \(n\)th particles is \((n-1)d-(n-2)\frac{Wa^2}{2T}\) where \(d\) is the vertical distance between the first and second particles and \(T\) is the horizontal tension of the string.
A uniform heavy sphere rests in contact with two parallel horizontal rods which are supported on a pair of fixed parallel horizontal rods perpendicular to the first pair and not in contact with the sphere. Prove that, if all the surfaces are equally rough and the system is on the point of motion, \[ (W+W')\sin(\beta-2\lambda)=W'\sin\beta, \] where \(W\) is the weight of the sphere, \(W'\) that of either of the movable rods, \(2\beta\) the angle subtended at the centre of the sphere by the chord joining the points of contact with the rods and \(\lambda\) the angle of friction.
Assuming the rods \(AB, BC, CD, DA\) in the framework of question 2 to be heavy and uniform while the weight of \(AC\) is negligible, apply the principle of virtual work to prove that the tension in \(AC\) is \(\frac{1}{2}\{ \frac{1}{2}W+W_1+W_2\}\operatorname{cosec}\theta\) where \(W\) is the total weight of the rods, \(W_1, W_2\) the weights suspended at \(B, C\), and \(\theta=\angle BAC\).
A train is running on a level track at a speed of 50 miles per hour. Find the brake resistance in pounds per ton necessary to stop the train in half a mile. Find also the distance in which the same brake resistance would bring the train to rest when ascending a gradient of 1 in 120.
Two rings of masses \(M, m\) (\(1
A wedge of mass \(M\), whose faces are inclined at angles \(\alpha, \beta\) to the horizontal, is free to move on a horizontal plane. Particles of masses \(m_1, m_2\) are placed on its two faces respectively. Taking all the surfaces to be smooth, prove that the particle \(m_1\) will remain at rest relative to the wedge provided that \[ (M+m_1)\tan(\alpha+\beta)=(M+m_1+m_2)\tan\beta. \]
A heavy particle is projected from a point with velocity \(V\) so as to pass through another point at a distance \(r\). Prove that there are in general two possible directions of projection, and that the times of flight are roots of the equation \[ g^2t^4-2(V^2+V_1^2)t^2+4r^2=0, \] where \(V_1\) is the velocity at the second point.
Find the time of a small oscillation of a simple pendulum; find also the pressure on the point of suspension in any position when the oscillation is not necessarily small. \(ABC\) is a light framework in the form of an isosceles triangle, having \(B=C=30^\circ\) and \(AB=9\) in. Equal masses are fixed at \(B, C\) and the system is suspended from \(A\). Prove that the time of a small oscillation is nearly 1.36 sec.