A water-trough for cattle is made by putting semicircular ends on to a hollow half-cylinder of length \(l\) and radius \(r\). The sheeting from which the trough is made has weight \(W\) per unit area. If the trough is filled to the brim with water of weight \(w\) per unit volume, how far below the surface of the water will the centre of gravity of the full trough be?
Four uniform bars \(AB, BC, CD, DA\) of length \(a\) and weights \(w, 2w, w, 2w\) respectively are freely jointed at \(A, B, C\) and \(D\). \(A\) and \(C\) are connected by a light inextensible string of length \(l < 2a\) and the whole framework is suspended from \(A\). Find the tension in the string.
A uniform flexible heavy string is suspended from each end and hangs freely under gravity. Show that the intrinsic equation of the curve in which it hangs is \(s=c\tan\psi\) (where \(s, \psi\) and \(c\) should be defined), and deduce that the Cartesian equation of the curve, referred to suitable axes, may be put in the parametric form \[ x=c\log(\tan\psi+\sec\psi), \quad y=c\sec\psi. \] The ends of a uniform flexible string of weight \(W\) are attached to light rings which can slide along a fixed rough horizontal rod. A weight \(W\) is suspended from the mid-point of the string. Show that the ratio of the least possible sag at the middle of the string to the length of the string is \[ \frac{1}{2}\sqrt{(1+\mu^2)} - \frac{1}{2}\sqrt{(1+4\mu^2)}, \] where \(\mu\) is the coefficient of friction between each ring and the rod.
A light ladder of length \(l\) rests at an angle of 45\(^\circ\) to the vertical, with its foot on the ground and its head against a vertical wall. The coefficients of friction at the two ends of the ladder are both \(\mu\) (\(<1\)). A man walks very slowly up the ladder. Show that he can go a distance \(l(\mu+\mu^2)/(1+\mu^2)\) before it starts to slip. Discuss briefly whether he could have gone further by varying his speed.
Two light elastic strings \(AB, BC\) are connected at \(B\) and attached to points \(A\) and \(C\) respectively which are at the same level and distance \(2l\) apart. The strings each have unstretched length \(l\) and modulus of elasticity \(\lambda\). A weight \(w\) is placed at \(B\). If the weight remains in equilibrium when \(AB=BC=2l\), show that \(w=\lambda/\sqrt{3}\). The weight is given a small vertical displacement from its equilibrium position and then released. Find the period of the small oscillations which it performs.
A railway engine of weight \(W\) lbs. is moving initially at a steady velocity \(v_0\) under no external forces. It begins to pick up water at a rate of \(w\) lbs. per unit length travelled. How long will it take to pick up its own weight of water?
Two small spheres \(A\) and \(B\) of masses \(3m\) and \(m\) respectively lie on a horizontal table, so that \(B\) lies between \(A\) and a perfectly elastic barrier perpendicular to the line of centres of \(A\) and \(B\). Initially \(B\) is at rest and \(A\) is projected towards \(B\) along the line of centres. If the coefficient of restitution between the spheres is \(\frac{1}{2}\), show that there will be exactly three collisions.
A ball of unit mass is thrown vertically upwards with velocity \(u\), and is subject to a resistance of magnitude \(k\) times the velocity. Show that it comes to rest after a time \(\frac{1}{k}\log(1+\frac{ku}{g})\) has elapsed, and find the height above the point of projection at that instant. It is desired to throw the ball to a height \(h\). Show that the least velocity required to achieve this is approximately \[ u = u_0\left(1+\frac{k}{3}\sqrt{\frac{2h}{g}}\right), \] where \(u_0\) is the corresponding minimum velocity in the absence of any resistance, and where \(k\) is so small that powers of \(k\sqrt{h/g}\) above the first may be ignored.
The position of a point \(P\) in a plane is specified by its distance \(r\) from a fixed point \(O\) of the plane, and by the angle \(\theta\) between \(OP\) and a fixed line of the plane through \(O\). Obtain expressions for the components along and perpendicular to \(OP\) of the velocity and the acceleration of \(P\), in terms of \(r, \theta\) and their derivatives with respect to the time. A particle \(P\) moves in a plane under the influence of a force of magnitude \(\mu/r^2\) per unit mass, directed towards \(O\). Show that \[ \ddot{\theta} = h/r^2, \quad \dot{r}^2 = C+\mu/r-\frac{1}{2}h^2/r^2, \] where \(h\) and \(C\) are certain constants. Deduce that the orbit of \(P\) will be a circle centre \(O\) if and only if \(\mu^2+2h^2C=0\), and find its radius in this case.
A thin uniform rod \(AB\) of mass \(m\) and length \(2l\) is smoothly hinged to a fixed point at \(A\), and is free to swing in a certain vertical plane. A small imperfectly elastic peg is fixed at a distance \(\frac{3}{4}l\) vertically below \(A\). The rod is released from rest in a horizontal position, and swings freely until it impinges on the peg, when it rebounds and next comes instantaneously to rest when making an angle \(\alpha\) with the vertical. Show that the impulse at the hinge is zero, and find that at the peg.