A circle touches the rectangular hyperbola \(x^2 - y^2 = a^2\) in two real points. Show that the circle belongs to one of the two families
A regular tetrahedron, with edges of length \(a\), is inscribed in a sphere of radius \(R\). Find the value of the ratio \(a/R\).
A river has parallel banks distance \(2h\) ft. apart. The velocity of the stream vanishes at the banks and increases linearly to a maximum value \(u_0\) ft./sec. at the centre. A swimmer who swims at \(v_0\) ft./sec. in still water crosses the river. How long does it take him if he crosses as quickly as possible, and how far downstream from his starting point does he finish? Show that, if \(v_0 > u_0\) and he swims in such a way that he is always moving towards the point immediately opposite his starting position, his travel time is $$\frac{2h}{u_0} \sin^{-1} \frac{u_0}{v_0} \text{ seconds.}$$
A uniform rod of length \(l\) has a ring at one end which slides on a smooth vertical wire. A smooth cylinder of radius \(a\) has its axis horizontal and at a distance \(b\) (\(b > a\)) from the wire. The rod rests in tangential contact with the cylinder in a plane perpendicular to its generators. Find a relation between \(l\) and the angle \(\theta\) which the rod makes with the wire. Deduce that if \(l > 2b\) there exist two positions of equilibrium for all values of \(a\).
Two equal rough circular cylinders of weight \(W_1\) touch one another along a horizontal generator and both rest upon a rough horizontal plane. A third cylinder of the same radius and of weight \(W_2\) rests above the first two, also touching each along a generator. The coefficient of friction between the table and any cylinder is \(\mu_1\), and that between the upper and lower cylinders is \(\mu_2\). Show that, for the cylinders to remain in equilibrium, $$\mu_1 \geqslant (2-\sqrt{3})W_2/(2W_1 + W_2), \quad \mu_2 \geqslant 2-\sqrt{3}.$$
Two particles collide and coalesce. Show that it is impossible for mass, momentum, and kinetic energy all to be conserved in such a collision. In particular, if mass and momentum are conserved, find an expression for the energy loss in terms of the masses \(m_1\), \(m_2\) of the particles and their velocities \((u_1, v_1)\), \((u_2, v_2)\) in the common plane of their trajectories. Show further that, if the particle velocities and the total mass are prescribed, the energy loss is greatest if the particles have equal mass.
A spherical star of initial mass \(M_0\) and radius \(a\) moving with velocity \(v_0\) enters a cloud which is at rest. The cloud has density \(\rho\) and thickness \(b\) in the direction of the star's motion. All the particles of the cloud struck by the star are absorbed by it without changing its radius. What is the velocity of the star when it leaves the cloud and how long does it take to cross it? (It may be assumed that \(a\) is small compared with \(b\).)
Two particles of equal mass are joined by a light inextensible string of length \(\pi a/3\). Initially they rest in equilibrium with the string across the top of a smooth circular cylinder of radius \(a\). The particles are then slightly disturbed from rest, the string remaining taut. Find the position of the particles when the first one leaves the cylinder.
\(A\), \(B\) and \(C\) are three equal particles attached to a light inextensible string at equal intervals \(a\). The system is placed on a smooth horizontal table with the three particles in a straight line. \(B\) is suddenly started moving with velocity \(v\) perpendicular to the string. Show that, until the first impact, the angular velocity of \(AB\) is given by \(v/a(2 + \cos^2 \theta)\), where \(\theta\) is the angle \(ABC\).
Show that the energy stored within an elastic string, of natural length \(L\) and modulus \(E\), when stretched to a length \(L + l\), is \(\frac{1}{2}El^2/L\). A mass \(m\) is attached by two elastic strings, of natural lengths \(L_1\) and \(L_2\) and moduli \(E_1\) and \(E_2\) respectively, between two fixed points a distance \(L_3\) apart on a smooth horizontal table, where \(L_3 > L_1 + L_2\). What is the stored elastic energy when the system is at rest? Show that if the mass is displaced slightly towards either fixed point the period of small oscillations is $$2\pi \sqrt{\frac{mL_1 L_2}{E_1 L_2 + E_2 L_1}}.$$