A shot from a gun is observed to fall a distance \(d\) short of its target, which is well within range and at the same level as the gun, where \(d\) is small compared to the distance of the target. Show that the elevation of the gun must be increased by \(gd/(2V^2\cos 2\beta)\), approximately, where \(V\) is the muzzle velocity and \(\beta\) is the elevation at which the gun was first fired.
A catapult is formed by holding a particle of mass \(m\) against the mid-point of a light elastic string of natural length \(2l\) and modulus \(\lambda\), whose ends are fixed at a distance \(2l\) apart, and then pulling back horizontally a distance \(\frac34l\). The whole system lies in contact with a smooth horizontal table. Show that when the particle is released it attains a final velocity of \[ \sqrt{(\lambda l/8m)}. \] What difference, if any, does it make if the catapult is made from two elastic strings of length \(l\) and modulus \(\lambda\) joined end to end by a non-elastic connection whose length and mass may be neglected?
A particle of mass \(m\) is placed at the top of the inclined face of a smooth wedge of mass \(M\), height \(h\) and angle \(\alpha\), which rests on a smooth horizontal plane, and is then let go. The particle slides down the face of the wedge and is caught by a small hole at the bottom of the wedge and remains there. Find the final position and state of the system.
A train of mass \(M\) travels along a horizontal track; the resistance to motion is \(kv^2\), where \(v\) is the velocity of the train. Show that, if the engine is assumed to work with constant power \(P\) and to start from rest, then the velocity of the train never exceeds \(\sqrt[3]{(P/k)}\), and that when the velocity is half this amount the distance gone is \((M/3k)\log(8/7)\).
A man of mass \(M\) carrying a hammer of mass \(m\) stands on the circumference of a light circular horizontal platform of radius \(a\) which is free to rotate about its centre. The man swings the hammer in a horizontal circle of radius \(b\) with himself as centre, where \(b< a\). The radius to the man makes an angle \(\phi\) with a fixed direction and the line from the man to the hammer makes an angle \(\theta\) with the same fixed direction. Show that \[ (M+m)a\dot{\phi}^2 + m[b^2\dot{\theta} + ab(\dot{\theta}+\dot{\phi})\cos(\theta-\phi)] = 0. \]
A circle \(A\) of radius \(a\) (\(a>b\)) rotates with angular velocity \(\omega\) about its centre \(O\) which is fixed. A circle \(C\) of radius \(\frac{1}{2}(a-b)\) touches the circle \(A\) and a fixed circle \(B\) of radius \(b\) and centre \(O\). If no slipping occurs between \(A\) and \(C\) or between \(B\) and \(C\), show that the angular velocity of \(C\) is \(\displaystyle\frac{a\omega}{a-b}\) and that the angular velocity of the line of centres is \(\displaystyle\frac{a\omega}{a+b}\).
A chain of length \(l\) lies in the smooth horizontal arm of an \(\Gamma\)-shaped tube. The other arm of the tube hangs vertically downwards and the corner of the tube is rounded. A fraction, \(k\), of the chain hangs over the corner and dips into the vertical arm. Show that the chain will be free of the horizontal arm in time \[ \sqrt{\frac{l}{g}} \log\left(\frac{1+\sqrt{(1-k^2)}}{k}\right). \] If the chain breaks when the tension is \(\frac{4}{25}\) of its weight, show that it will break when \(\frac{3}{5}\) of the chain is past the corner.
Prove that a coplanar system of forces may be reduced to a force through an assigned point and a couple. Show also that, in general, the system is equivalent to a single force. A square lamina \(ABCD\) lies on a smooth horizontal table and is subject to a force \(F\) acting at \(A\) along \(DA\) produced, a force \(2F\) at \(B\) along \(AB\) produced, and a force \(2F\) at \(C\) towards \(B\). Show that if the lamina is freely hinged at \(D\) it will not move. Find the force on the hinge.
A uniform ladder of weight \(w\) rests with one end on the ground and with the other against a vertical wall, its angle of inclination being \(45^\circ\). The coefficient of friction at each end is \(\frac{1}{2}\). A man of weight \(4w\) begins to climb the ladder. Show that slipping will commence when he has covered three-eighths of the length of the ladder.
A closed rectangular box is made of thin uniform sheet, its base being a square of side \(a\) and its height \(\frac{3}{2}a\). The base is made of double thickness of sheet and the rest of the box is made of sheet of single thickness. The box stands on a perfectly rough inclined plane four of its edges being parallel to the line of greatest slope. A horizontal force equal to one-third of the weight of the box is applied along the perpendicular bisector of the highest edge so as to tend to topple the box down the slope. Show that if the force is just able to topple the box then the inclination of the plane to the horizontal is \(\tan^{-1}\frac{5}{8}\).