A curve whose polar equation is \(f(r, \theta)=0\) has pedal equation \(F(r,p)=0\). Prove that the curve whose polar equation is \(f(r^n, n\theta)=0\) has pedal equation \(F(r^n, p r^{n-1})=0\).
A ladder, inclined at \(30^\circ\) to the vertical, leans against a vertical wall. The centre of gravity is half-way up the ladder, and the upper end can slide smoothly up and down the wall. Show that the ladder will not slip if the angle of friction between the ladder and the ground is greater than about \(16^\circ\) 6'. If the angle of friction is \(30^\circ\) and the weight of the ladder is \(W\), find the magnitude and direction of the least force which, if applied to the foot, will cause the ladder to slide outwards.
A number of particles, all of the same weight, are attached to a light string at points \(P_0\), \(P_1\), \(P_2\), \dots. If the horizontal intervals between these points are all equal, prove that the tensions in \(P_0P_1\), \(P_1P_2\), \dots, are proportional to their lengths. Show that \(P_0\), \(P_1\), \(P_2\), \dots, lie on a parabola with a vertical axis.
A lamina is displaced in its own plane. Prove that the displacement is either a rotation about some point or a translation in which every point receives the same displacement. A symmetrical three-legged stool of weight \(W\) stands upon the ground. The coefficient of friction between the feet of the stool and the ground is \(\mu\). A gradually increasing force \(P\) is applied to one of the feet \(A\) in a direction parallel to the join of the other two feet. Assuming that equilibrium is broken by the stool rotating about some point on the diameter through \(A\), determine that point, and find the value of \(P\) for which this takes place.
Define shearing stress and bending moment, and explain with the aid of a clear diagram what conventions of sign you adopt. A uniform horizontal straight rod of length \(2l\) and weight \(w\) per unit length rests sym- metrically on two smooth supports distant \(r\) from its centre; it carries a concentrated load \(2W\) at the mid-point. Write down expressions for the shearing stress and bending moment at a distance \(x\) from the centre. If \(W=wl\) and \(l=2r\), find the points at which the bending moment vanishes.
A point is moving with simple harmonic motion, of period \(2\pi/n\) and amplitude \(a\), in a straight line. If at any instant the distance of the point from its mean position is \(x\), show that the speed of the point is \(n\sqrt{a^2 - x^2}\). A mass \(m\) hangs at rest at the lower end of a light elastic string, of unstretched length \(l\) and modulus of elasticity \(\lambda\). A second mass \(m\), moving vertically upwards with velocity \(U\), impinges on the first mass and coalesces with it. Show that in the subsequent motion the string remains taut provided that \(\lambda U^2 < 6mlg^2\).
A projectile, of mass \(m\), is fired horizontally from a gun, of mass \(M\), which is free to recoil. The length of the barrel is \(l\), and the force \(P\) exerted by the propellant has the constant value \(P_0\) until the projectile has travelled a length \(\frac{3}{4}l\) of the barrel, after which it steadily decreases from \(P_0\) to zero so that \(P\) is proportional to the distance from the muzzle. Show that the square of the velocity of the projectile just after leaving the muzzle is \[ \frac{7MlP_0}{4m (M+m)}. \]
A shell explodes at a height \(h\) above level ground, and fragments are assumed to fly in all directions with the same speed \(V\). Show that all the fragments reach the ground on or within a circle of radius \(V\sqrt{(V^2 + 2gh)}/g\).
A chain, whose weight per unit length may be taken as constant, is of length \(l\) and weight \(W\). Initially it is held at rest by the upper end, with the lower end just touching a smooth horizontal inelastic table. If the chain is allowed to fall freely, show that when a length \(x (< l)\) lies on the table the total reaction on the table is \(3Wx/l\).
A rigid body is free to rotate about a fixed axis; show that the angular acceleration is \(G/I\), where \(I\) is the moment of inertia about the axis, and \(G\) is the moment of the applied forces about the axis. Two wheels, \(A\), \(B\), of radii \(r_A\), \(r_B\) and of moments of inertia \(I_A\), \(I_B\) respectively, are mounted on parallel axles; a light non-slipping belt passes round both wheels. A couple \(G\) is applied to wheel \(A\) about its axis. Find the angular acceleration of each wheel, and the difference in the tensions in the free portions of the belt.