Shew that a parallelogram of freely jointed rods is in equilibrium under forces in its plane at the joints, if the polygon of the forces is closed and has its diagonals parallel to the rods. Shew also that the form is stable, supposing one joint fixed and the forces to retain their magnitudes and directions in space, provided the sums of the tensions in opposite rods are both positive, a thrust being reckoned negative.
A particle \(P\) is moving under the law of acceleration \(n^2.OP\) towards a fixed point \(O\): initially the particle is moving away from \(O\) with velocity \(v_0\), and its distance from \(O\) is equal to \(x_0\). Shew that the particle next comes to rest at a distance \(\sqrt{(x_0^2+v_0^2/n^2)}\) from \(O\): and calculate the time which elapses. An elastic string has one end fixed at \(A\), and at the other end is fastened a particle heavy enough to stretch the string (statically) to twice its natural length \(a\). The particle is dropped from rest at \(A\); what time will elapse before the particle next comes to rest?
A bullet of mass \(m\) is fired into a block of wood of mass \(M\), which hangs by vertical cords of equal length, the other ends of the cords being fastened to fixed points on the same level. The bullet penetrates a distance \(a\) horizontally into the block and in the subsequent motion the block rises through a height \(h\). Calculate the velocity with which the bullet strikes the block, and shew that the average resistance to penetration is equal to \(gM(1+M/m)h/a\).
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