A uniform rod of length \(2a\) and mass \(m\) is pivoted at a point distant \(h\) from its centre. If \(\theta\) is the inclination of the rod to the vertical, write down the equation which gives \(\ddot{\theta}\). \par By considering the moment about the pivot of the forces on the part of the rod below the pivot, shew that there must be a bending couple in the rod at the pivot equal to \[ \frac{mg\sin\theta(a^2-h^2)^2}{4a(3h^2+a^2)}, \] when \(\theta\) is the inclination of the rod to the vertical.
The ends of a light string are attached to two smooth rings of weights \(w\) and \(w'\), and the string carries a third smooth ring of weight \(W\) which is free to slide on the string. The rings \(w\) and \(w'\) slide on two smooth rods in the same vertical plane and inclined at angles \(\alpha\) and \(\beta\) to the vertical. If equilibrium is possible with the ring \(W\) hanging down on the string between \(w\) and \(w'\), show that \begin{equation} (W+2w')\tan\alpha = (W+2w)\tan\beta, \tag{1} \end{equation} and that the two parts of the string are inclined at the same angle \(\phi\) to the vertical, where \begin{equation} (W+2w)\cot\phi = W\tan\alpha. \tag{2} \end{equation} Indicate what positions of equilibrium exist if the relation (1) is not satisfied.
A smooth hollow circular cylinder of radius \(R\) is fixed with its axis horizontal, and three smaller smooth uniform circular cylinders each of radius \(r\) and weight \(w\) are placed inside, two being in contact symmetrically placed about the lowest generator of the fixed cylinder, and the third resting on top of these two. Show that the position is one of equilibrium only if \[ R < r(1+2\sqrt{7}). \]
Three equal uniform rods, each of weight \(W\) and length \(l\), are freely hinged together at one end A. The other ends of the rods rest on a horizontal plane so that a tripod is formed with vertex A.
A uniform solid hemisphere rests with its base in contact with a rough plane inclined at an angle \(\alpha\) to the horizontal, and a force P is applied to the hemisphere at the point farthest from the plane, its line of action being parallel to a line of greatest slope up the plane. If P is gradually increased from the minimum value required to prevent the hemisphere from slipping down the plane, show that the equilibrium is broken by tilting or slipping according as the coefficient of friction is greater than or less than \(1-\frac{8}{3}\tan\alpha\). \par When equilibrium is broken by slipping, determine the line of action of the reaction exerted by the plane on the hemisphere.
The gunner in a moving tank aiming to hit a moving enemy tank must point his gun in advance of the enemy tank to allow for movement. The angle between the bore of the gun and the line joining the two vehicles is called the aim-off. A tank A, travelling due east, is 500 yards due north of a tank B which is travelling due north, both tanks moving steadily at 20 miles per hour. The tank B fires at A with a gun whose muzzle velocity is 3000 ft. per sec. Assuming that the bullet goes in a straight line with constant speed, show that the aim-off required increases to a maximum which is attained at the instant when each tank has travelled 250 yards, and that this maximum aim-off is approximately 48'.
An anti-tank gun fires a projectile weighing 2 lb. with a muzzle velocity of 3000 ft. per sec. The shell travels 9 ft. before leaving the muzzle, and the rate of burning of the charge is such that the acceleration of the shell is sensibly uniform throughout its passage to the muzzle. Show that the (sensibly constant) force exerted on the base of the shell by the gases is approximately \(3 \times 10^4\) pounds weight. \par The gun itself weighs 2 cwt. and is checked in its recoil by springs which allow a recoil of 1 ft. Show that at the fullest point of recoil the force exerted by the springs is approximately \(5 \times 10^3\) pounds weight. (It may be assumed that the gun does not recoil appreciably while the shell is in the muzzle.)
Neglecting air resistance, show that, for a projectile fired under gravity, the maximum range on a horizontal plane is \(V^2/g\), where \(V\) is the muzzle velocity and \(g\) is the acceleration of a freely falling body. For a 2-in. mortar whose maximum range is 530 yd., show that the muzzle velocity is approximately 225 ft. per sec. Given that \(\tan 36^\circ 52' = \frac{3}{4}\), show that the range corresponding to an elevation of \(36^\circ 52'\) is 509 yd. What other elevation gives the same range? \par An approximate allowance for the effect of a head wind of velocity \(w\) is given by assuming that the horizontal component of the velocity of the projectile suffers a constant retardation \(kw\). Two successive shots are fired against a head wind from the mortar at the two (low angle and high angle) elevations which in still air would give a range of 509 yd. If \(w\) is such that \(kw = 2 \text{ ft./(sec.)}^2\), show that the actual ranges are 485 yd. and 467 yd.
A particle of mass \(m\) is constrained to move on a smooth wire in the shape of a parabola whose axis is vertical and whose vertex is upwards. The particle is projected from the vertex with velocity \(u\). Show that the pressure on the wire at any point is \[ \frac{m}{\rho}(gl-u^2), \] where \(2l\) is the latus rectum, and \(\rho\) is the radius of curvature.
A uniform rod of mass \(m\) and length \(2a\) is supported horizontally by two elastic strings, each of natural length \(l\) and modulus of elasticity \(\lambda\), which are attached to a fixed point vertically above the middle of the rod. In equilibrium the strings make an angle \(\theta\) with the vertical. Show that the period of small oscillations in which the rod remains horizontal is \(2\pi/n\), where \[ n^2 = \frac{g}{a}\frac{a-l\sin^3\theta}{a-l\sin\theta}\tan\theta. \]