A uniform rectangular lamina rests with its plane vertical on two fixed smooth pegs. If one diagonal is parallel to the straight line joining the pegs, prove that the other diagonal is vertical.
On a fixed circular wire (radius \(r\)) in a vertical plane slide two small smooth rings, each of weight \(W\). The rings are joined by a light inextensible string of length \(2a\) (\(<2r\)) on which slides a small smooth ring of weight \(P\). Prove that for equilibrium either both parts of the string are vertical, or else \(P\) is at a distance from the centre of the wire equal to \(\{(r^2-a^2)W/(W+P)\}^{\frac{1}{2}}\).
A straight uniform pole \(AB\) leans against a vertical wall. The lower end \(A\) is on the horizontal ground \(a\) feet from the wall; the upper end \(B\) is on the wall, \(b\) feet above the ground and \(c\) feet to one side of the vertical plane through \(A\) that is perpendicular to the wall. Assuming that the ground is rough enough to prevent slipping at \(A\), prove that to prevent slipping at \(B\) the coefficient of friction between the pole and the wall must not be less than \[ \frac{c}{ab}\sqrt{b^2+c^2}. \]
Two masses \(M\) and \(m\), connected by a light spring obeying Hooke's law, fall in a vertical line with the spring unstressed until \(M\) strikes an inelastic horizontal table. Prove that \(M\) will after an interval rise from the table if the distance through which \(M\) has fallen exceeds \(l\left(1+\frac{M}{2m}\right)\), where \(l\) is the extension that would be produced in the spring by a force equal to the weight of \(M\).
A heavy particle \(P\) is attached by two unequal light inextensible strings to fixed points \(A, B\) in the same horizontal line, and is projected so as just to describe a vertical circle. When \(P\) is in its lowest position the string \(PB\) breaks, and \(P\) then describes a horizontal circle. Prove that the angle \(PAB\) is \(\cos^{-1}\frac{1}{2\sqrt{3}}\). Prove also that, if the tension of the string \(PA\) is unchanged when the string \(PB\) breaks, the angle \(APB\) is a right angle.
A railway wagon of mass 21 tons is shunted on to a siding and reaches a hydraulic buffer at a speed of 8 feet per second. This buffer is such that it exerts a constant force of 35 tons weight while being pushed in, but exerts only a negligible force while returning. The wagon buffer springs obey Hooke's law and require a total force of 7 tons weight to compress them both 1 inch. Prove that the wagon moves 9.7 inches after striking the buffer before coming to rest, and that it leaves the buffer at about 4.7 feet per second.
A uniform rope of length 5 feet and mass 5 lb. is placed over a small rough fixed horizontal peg so that the rope hangs vertically on both sides of the peg. A mass of 35 lb. is attached to one end of the rope, and initially is close to the peg, when the system is released from rest. Owing to friction the ratio of the tensions on the two sides of the peg is constant and equal to 5. Find the initial acceleration, and prove that the speed attained when the other end of the rope reaches the peg is about 13 feet per second. [\(\log_e 1.5 = 0.405\).]
If \[ y = (x+\sqrt{1+x^2})^m, \] prove that \[ (1+x^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx} = m^2y, \] and find the first three terms in the expansion of \(y\) in a series of ascending powers of \(x\). If \(z^2t = e^{\frac{x^2}{2kt}}\), where \(k\) is a constant, prove that \[ k\frac{\partial^2 z}{\partial x^2} = \frac{\partial z}{\partial t}. \]
\(C\) is the centre and \(P\) a given point (\(CP=b\)) on a spoke of a wheel of radius \(a\) that rolls along a straight line on the horizontal ground, all the motion being in a vertical plane. For the trochoidal locus of \(P\), prove that the curvature is numerically equal to \(\frac{b}{(a-b)^2}\) at the highest point, \(\frac{b}{(a+b)^2}\) at the lowest point, and zero when \(APC\) is a right angle, where \(A\) is the point of contact of the rolling circle with the ground.
(i) For the curve \(y^2 = x(x-1)(2-x)\), prove that the greatest length and breadth of the loop, measured parallel to the axes of \(x\) and \(y\), are 1 and \(\frac{2}{3\sqrt{3}}\). (ii) Prove that the area included between the curve \(y^2(1-x)=x^3\) and its asymptote is \(\frac{3}{4}\pi\).