A bead is made by boring a cylindrical hole of radius \(r\) through a uniform sphere of radius \(R\), the axis of the hole passing through the centre of the bead. The bead has radius of gyration \(k_1\) about the axis and \(k_2\) about a diameter perpendicular to the axis. Show that $$k_1^2 = \frac{1}{2}(2R^2 + 3r^2) = k_2^2 + \frac{1}{2}r^2.$$
A plane framework consists of five uniform heavy rods \(AB\), \(BC\), \(CD\), \(DA\), \(AO\), smoothly hinged at \(A\), \(B\), \(C\) and \(D\), and two light elastic strings \(BO\) and \(DO\); \(O\) lies outside the rhombus \(ABCD\). Each rod is of length \(l\) and weight \(W\), and each string has natural length \(l\) and modulus \(6W\). The framework is suspended freely from \(O\). Show that there is a position of equilibrium in addition to that in which the rods are all vertical, and examine the stability of these two positions of equilibrium. Find the reaction at \(C\) between the rods \(BC\) and \(CD\) when the framework hangs in the equilibrium position in which the rods are not all vertical.
An inextensible flexible string \(AB\), of uniform weight \(w\) per unit length, hangs freely with its ends fixed. The chord \(AB\) is not vertical, and the tangent at the general point \(P\) of the string makes an angle \(\psi\) with the horizontal. Show that $$\frac{s-\alpha}{\tan \psi} = \frac{y-\beta}{\sec \psi} = \frac{T}{w \sec \psi} = c,$$ where \(s\) is the length of the arc \(AP\), \(y\) is the height of \(P\) above \(A\), \(T\) is the tension at \(P\), and \(\alpha\), \(\beta\), \(c\) are constants. One end of a uniform inextensible flexible string, of length \(l\) and total weight \(W\), is fixed at ground level, and the other is attached to a flying kite of weight \(4W\). The string at ground level makes an angle \(\tan^{-1} \frac{2u}{3}\) with the horizontal, and the resultant force exerted on the kite by the wind makes an angle \(\tan^{-1} \frac{2u}{3}\) with the horizontal. Find: (i) the tension in the string at ground level, (ii) the height of the kite above the ground. [The effect of the wind on the string is to be neglected.]
A rigid beam of length \(l\) and weight \(W\) has the shape of a frustum of a slender right-circular cone, the radii of the cross sections at its ends being \(r\) and \(2r\). The material of the beam is of constant density. Find the shearing force and bending moment at the general point of the beam when it is supported with its axis horizontal by a peg at each end. Show also that the maximum numerical value of the bending moment in this case is about \(32\%\) of what it would be if the beam were held in a horizontal position by a clamp at the thick end, the thin end being unsupported. [\(\sqrt{30} = 3.107\) approx.]
A uniform heavy rod is in equilibrium with one end resting on a fixed horizontal plane and the other supported by a vertical string; the rod makes an angle of \(45^\circ\) with the vertical. The string is then cut. Show that, so long as the coefficient of friction \(\mu\) between the rod and the plane does not exceed a certain critical value \(\mu_0\), the normal reaction of the plane on the rod immediately after the string is cut is \(\frac{4}{5}(\sqrt{5}-3\mu)\) times its value before the string is cut. Find \(\mu_0\), and determine the corresponding ratio in the case \(\mu > \mu_0\).
The maximum range of a gun on level ground is \(r\). Show that the trajectory of a shell, when fired at any angle of elevation, is a parabola whose directrix is horizontal and at a height \(\frac{1}{2}r\) above the ground. Show also that the envelope of all possible trajectories in a given vertical plane through the gun is a parabola whose focus is at the gun and whose directrix is horizontal and at a height \(r\) above the ground. The gun is mounted at the top of a vertical cliff at a height \(h\) above the sea. A ship at sea has a gun whose maximum range on a horizontal plane is \(R\). Show that it is possible for the ship to engage the cliff-top gun while remaining out of its range if \(R > r + 2h\).
Three equal smooth spheres, with coefficient of restitution \(e\), lie in a straight line on a smooth table. One of the outer spheres is projected in the direction of the other two. Show that there will be subsequently two collisions if \(e = 1\) and three collisions if $$\frac{3-2\sqrt{2}}{2} \leq e < 1.$$ Describe what you would expect to happen as \(e \to 0\).
A toy motor car consists of a body of mass \(4m\) and four road wheels, each of mass \(m\), radius \(a\), and moment of inertia \(\frac{1}{2}ma^2\). A second motor car is similar except for the addition of a flywheel of mass \(m\) and moment of inertia \(\frac{1}{2}ma^2\) geared to the road wheels in the ratio \(\sqrt{(23)}:1\). Show that, for given initial velocity, the second motor car will travel twice as far as the first up an inclined rough plane before coming to rest.
A thin uniform rod of mass \(m\) and length \(2a\) can turn freely about one end which is fixed, and a circular disk of mass \(12m\) and radius \(\frac{1}{4}a\) can be clamped to the rod so that its centre is on the rod. Show that, for oscillations in which the plane of the disk remains vertical, the length of the equivalent simple pendulum lies between \(2a\) and \(\frac{8}{3}a\).
A rocket in rectilinear motion is propelled by ejecting all the products of combustion of the fuel from the tail at a constant rate and at a constant velocity relative to the rocket. Show that, for a given initial total mass \(M\), the final kinetic energy of the rocket is greatest when the initial mass of fuel is \((1-e^{-2})M\).