A continuous flexible chain (not necessarily uniform) hangs under gravity between two points so that its lowest point is at neither end. Show that if the shape of curve in which it hangs is known, the value at any point of \(w\) the weight per unit length is proportional to \(\sec^2\psi/\rho\), with the usual notation. Prove that if the curve is part of a section of a cycloid whose line of cusps is horizontal, the relation between \(w\) and \(s\) the length of arc measured along the curve from the lowest point is of the form \[ w_0^2 = w^2\left(1 - \frac{s^2}{a_0^2}\right)^3, \] where \(w_0\) and \(a\) are constants. [The cycloid is given by coordinates \(x\) and \(y\) referred to axes along the tangent and normal at the point of least curvature in terms of a parameter \(\lambda\) by the equations \(x=a(\lambda+\sin\lambda)\), \(y=a(1-\cos\lambda)\).]
A smooth uniform heavy sphere of weight \(W\) and radius \(a\) suspended from a point \(O\) by a light string of length \(l\) whose other end is attached to a point \(A\) of the spherical surface rests in stable equilibrium in contact with a smooth plane through \(O\) and inclined at an angle \(\alpha\) to the vertical. Prove that the tension \(T\) in the string is given by \[ T = W(l+a)\cos\alpha/[l(l+2a)]^{\frac{1}{2}}. \] Prove that if the sphere (still of uniform material) were hollow with an internal smooth concentric spherical cavity containing smooth weights free to move, the position of stable equilibrium is the same as before, and determine the tension in the string in terms of the weights of the bodies involved.
A heavy uniform rod \(AB\) is held in equilibrium at an inclination \(\alpha\) to the vertical with one end resting on a rough horizontal plane and the other end supported by the tension of a string inclined at \(\theta\) to the vertical. If \(\mu=\tan A\) is the coefficient of friction between the rod and the plane, prove that the greatest possible value of \(\theta\) is given by \(\cot\theta = \cot A \pm 2\cot\alpha\), where the alternate sign is chosen if the string and rod are inclined to the vertical in opposite senses. Explain what happens if \(\mu \ge \frac{1}{2}\tan\alpha\).
A small heavy sphere suspended from a fixed point \(O\) by a light elastic string will hang in equilibrium at a height \(a\) above a horizontal plane with the extension of the string from its natural length at the same value \(a\). The sphere is released from rest at a height \(b\) vertically above the plane where \(b>2a\). If \(e\) is the coefficient of restitution for impact with the plane, show that after \(n\) impacts the sphere comes instantaneously to rest at a height above the plane \(2a+e^{2n}(b-2a)\).
A particle is projected from a point \(O\) with velocity \(u\) at an angle \(\alpha\) to the horizontal and moves under gravity in a medium offering a resistance per unit mass of \(kv\), where \(v\) is the speed and \(k\) is a positive constant. Prove that there is a fixed direction in which the component of velocity is constant, and show that the value \(V\) of the component is related to the velocity \(u_0\) at the highest point of the path by the equations \[ V = \frac{g}{k}\cos\beta, \quad u_0 = \frac{g}{k}\cot\beta, \] where \(\beta\) is an acute angle. State the significance of the angle \(\beta\).
Find the radial and transverse components of acceleration of a point moving in a plane and whose position at any time is given by polar coordinates \(r, \theta\). Find the law of attraction to the origin for a particle projected with velocity \(V\) at the point \(r=a, \theta=0\), to describe the equiangular spiral \(r=ae^{\theta\cot\alpha}\).
A body free to rotate about an axis through its centre of mass has its motion controlled so that it can execute simple harmonic motion of period \(2\pi/n\) in the absence of friction. It is found that with a frictional couple proportional to the angular velocity the motion is still oscillatory but the ratio of successive angular displacements (regardless of sign) is the proper fraction \(\lambda\). Show that if released from rest with angular displacement \(\alpha\), the system first passes through the neutral position with angular velocity given by \(n\alpha\lambda^{\beta} e^{-\beta\tan\beta}\), where \(\beta\) is the acute angle given by the equation \(\pi\tan\beta = -\log_e\lambda\).
A bead is threaded on a rough wire bent in the form of a circle held fixed in a vertical plane. The coefficient of friction between the bead and wire is \(\frac{1}{\sqrt{2}}\). If \(u\) is the velocity of projection at the lowest point for the bead to come to rest at the level of the centre of the circle, and \(v\) is the velocity of projection downwards at this latter point for the bead to come to rest at the lowest point, prove that \[ u/v = \exp(\pi/2\sqrt{2}). \]
Prove that a particle moving under an inverse square law of attraction to a fixed centre of force \(S\) will describe a conic with \(S\) as a focus. Show that if when at a point \(P\) of the path the radial component of the velocity is destroyed, the subsequent path is a conic having \(P\) and \(P'\) at the ends of its major axis, where \(PSP'\) was a focal chord of the original path.
A thin circular hoop of radius \(a\) is made of non-uniform material so that the centre of mass is halfway between centre and circumference. The hoop moves in a fixed vertical plane and can roll without slipping on a rough horizontal plane. When the centre of mass is at the highest point the hoop is slightly disturbed from rest. Find an equation giving the value \(\theta\) of the angular velocity in terms of \(\theta\) the angular displacement of the radius through the centre of mass from its original position. Verify that the frictional force is zero when \(\cos\theta = -\frac{1}{4}\).