A system of \(n\) forces acts in the plane \(xOy\) at the points \((x_1,y_1), (x_2,y_2), \dots, (x_n,y_n)\); the components of the forces along the axes (which are orthogonal) are \[ (X_1,Y_1), (X_2,Y_2), \dots, (X_n,Y_n). \] Shew that the system can be reduced to two forces, one acting at a given point \(P\) and the other acting along a given straight line \(l\). Find these forces if \(P\) is the point \((h,k)\) and the equation to \(l\) is \(ax+by+c=0\).
A uniform ladder \(AB\) of length \(2l\) rests with one end \(A\) on the ground and the other end \(B\) in contact with a smooth vertical wall; the vertical plane through \(AB\) is perpendicular to the plane of the wall. If \(\mu\) is the coefficient of friction between the ladder and the ground, and \(\alpha\) the inclination of \(AB\) to the vertical, prove that \(\mu\) must exceed \(\frac{1}{2}\tan\alpha\). A force just large enough to move the ladder is now applied to the point of the ladder at a distance \(c\) from \(A\); the force acts away from, and perpendicular to, the wall. Find in what conditions the ladder (1) will rotate about \(A\), (2) will slip at \(A\) and \(B\).
A rectangular trap-door of weight \(W\) is free to rotate about two fixed smooth hinges attached to one side; the line joining the hinges makes an angle \(\alpha\) with the vertical. The door is kept in equilibrium in a position such that its plane makes an angle \(\beta\) with the vertical plane \(P\) through the hinges by a force \(F\) applied to the centre of gravity of the door in a direction perpendicular to the plane \(P\). Shew that \[ F = W \sin\alpha \tan\beta, \] and interpret this result if \(\beta \ge \pi/2\).
One end \(A\) of a uniform rod \(AB\) of length \(2a\) and weight \(W\) can turn freely about a fixed smooth hinge; the other end \(B\) is attached by a light elastic string of unstretched length \(a\) to a fixed support at the point vertically above, and distant \(4a\) from, \(A\). If the equilibrium of the vertical position of the rod with \(B\) above \(A\) is stable, find the minimum modulus of elasticity of the string.
A uniform wire hangs in equilibrium under gravity with its ends attached to two fixed supports on the same level at a distance \(2a\) apart; the length of the wire is such that the force exerted by it on either support is a minimum. If \(c\) is the parameter of the catenary in which the wire hangs and \(x = a/c\), shew that \[ \log_{10}(x+1) - \log_{10}(x-1) = 0.87x, \] approximately. Find an approximation to the root of this equation, and hence to the length of the wire.
A motor car stands at rest on a long straight horizontal road and a rifle is fired from the car, aimed along the road and with its axis inclined at an angle \(\alpha\) to the horizontal. The range of the bullet is found to be 1200 yards. The car now moves along the road at 10 miles per hour, and with the rifle fixed in the same position in the car a second shot is fired. Assuming that both bullets have the same muzzle velocity relative to the rifle, and neglecting air resistance and the height of the rifle above the road, shew that the second shot ranges 220 \((\tan\alpha)^{\frac{1}{2}}\) feet further than the first. What is the angle between the horizontal plane and the trajectory of the second bullet at the point of arrival of the bullet on the road?
A small spherical ball \(B\), of mass \(m\), hangs at rest under gravity at the end of a light inextensible string \(AB\) of length \(a\) which is fixed to a rigid support at \(A\). A second spherical ball of mass \(M\) impinges on the first with velocity \(V\), the velocity and the line of centres of the two spheres both being horizontal at the instant of impact. The string \(AB\) can support a tension of seven times the weight of \(B\), and the coefficient of restitution between the two balls is \(e\). Shew that, after the impact, \(B\) describes complete circles about \(A\) provided that \(V_0 < V \le V_1\), where \(V_0\) and \(V_1\) are certain fixed velocities. Determine \(V_0\) and \(V_1\) and explain what happens in the two cases \(V < V_0\) and \(V > V_1\).
A uniform fine chain of length \(3l/2\) and mass \(3ml/2\) hangs over a small smooth peg at a height \(l\) above a horizontal table. The chain is released from rest in the position in which it hangs in two vertical straight pieces with one end just touching the table. Shew that when the other end is leaving the peg, the force on the table is \[ mgl\left\{4\log\frac{3}{2} - \frac{1}{2}\right\}. \]
A heavy ring of mass \(m\) slides on a smooth vertical rod, and is attached to a light string which passes over a small light pulley at a distance \(a\) from the rod and is attached to a mass \(M(>m)\) which hangs freely. The system is released from rest when the string is taut, \(m\) level with the pulley, and \(M\) vertically beneath the pulley. Shew that the ring drops a distance \[ \frac{2Mma}{M^2-m^2} \] before coming to rest again. At this point the ring suddenly splits into two rings of equal mass, one of which falls freely while the string remains attached to the upper one. Shew that the upper ring rises to a height \[ \frac{2M^2-m^2}{4M^2-m^2}\frac{2Mma}{M^2-m^2} \] above the pulley before coming momentarily to rest.
A heavy particle slides in a light straight smooth tube which is pivoted at one end \(O\) and is free to rotate on a smooth horizontal table. The particle is attached to \(O\) by a light spring of such strength that when the tube is at rest the particle oscillates in the tube with period \(2\pi/n\). When the particle is at rest and the spring unstretched, the tube is suddenly given an angular velocity \(\omega\) about \(O\) and the system is then left free. If \(\omega=2n/\sqrt{3}\), shew that the greatest length of the spring is twice its natural length.