A heavy uniform rod \(AB\) is suspended in equilibrium under gravity by two equal inextensible light strings \(OA, OB\) attached to a fixed point \(O\). If one of the strings is suddenly cut, show that the tension in the other is instantaneously divided by the factor \(2+\frac{1}{2}\cot^2 OAB\).
A rigid body is in equilibrium under three forces. Show that their lines of action must be coplanar, and either parallel or concurrent. A uniform heavy bar hangs in equilibrium from two strings attached to its ends. The strings make angles \(\theta_1, \theta_2\) with the vertical and \(\phi_1, \phi_2\) respectively with the bar. Prove that \[ \sin\theta_1 \sin\phi_2 = \sin\theta_2 \sin\phi_1. \]
Four equal uniform rods, each of weight \(W\), are freely hinged to form a rhombus \(ABCD\), and a light rod of the same length joins \(BD\). The framework is suspended from \(A\), and a horizontal force is applied at \(C\) to maintain equilibrium with \(AD\) and \(BC\) vertical. Find the reaction at \(A\), and show that the thrust in \(BD\) is \(\frac{4}{3}W\).
A flexible cable of length \(2l\) and weight \(w\) per unit length will break if the tension exceeds \(\lambda wl\) (where \(\lambda>1\)). It is to be suspended by its ends from two points at the same horizontal level. Show that the distance between the points cannot exceed \[ l\sqrt{\lambda^2-1} \log\left(\frac{\lambda+1}{\lambda-1}\right). \]
A thin straight bar \(AB\) of length \(l\) is of variable density, having weight \(w(1+x/l)\) per unit length at distance \(x\) from the end \(A\). It is supported at \(A\) and \(B\) by pegs at the same horizontal level. Find the shearing force and bending moment at the general point of the bar, and show that the bending moment has its numerically greatest value at distance approximately \(0\cdot528l\) from \(A\).
A hollow cone (with base) is made out of thin material of uniform weight per unit area, and has semi-vertical angle \(\frac{1}{6}\pi\) and height \(h\). It is free to rotate about a horizontal axis through its vertex. Show that the length of the equivalent simple pendulum is \(27h/28\). [The thickness of the material may be neglected.]
A motor-car weighing 33 cwt. travels at a constant speed of 30 m.p.h. up a hill which is a mile long (measured along the road) and rises 625 ft. The engine works at a rate of 40 h.p. Find the resistance to the motion in lb. wt. Assuming that the resistance is proportional to the square of the speed, find the maximum speed at which the car can travel on a level road with the same rate of working. [1 h.p. = 550 ft. lb. per sec.; 1 cwt. = 112 lb.]
An aircraft is travelling along a straight line with velocity \(U\) and climbing at an angle \(\psi\) to the horizontal. A gun with muzzle velocity \(V\) is fired at it when the aircraft is immediately overhead at an altitude \(h\). Show that the shell cannot hit its target whatever the angle of projection unless \[ V^2 \ge 2gh + U^2 + \sqrt{(8gh)}U\sin\psi. \]
Two equal spheres, each of mass \(m\), collide, the coefficient of restitution being \(e\). Just before the impact the velocity of one sphere relative to the other has magnitude \(V\) and makes an angle \(\alpha\) with the line of centres. Show that the kinetic energy lost during the impact is \[ \frac{1}{2}(1-e^2)mV^2\cos^2\alpha. \] Show also that, if \(\tan^2\alpha=e\), the direction of the relative velocity is turned through a right angle and its magnitude is multiplied by \(e^{\frac{1}{2}}\).
A rod of length \(2l\) and mass \(m\) has a small ring at one end, which is free to slide along a smooth fixed horizontal wire. The rod is released from rest lying along the wire. Show that when it makes an angle \(\theta\) with the downward vertical, \[ (1+3\sin^2\theta)\dot{\theta}^2 = 6gl^{-1}\cos\theta, \] and find the reaction on the wire.