Find the equation of the two straight lines joining the origin with the points of intersection of the straight line \(lx+my=1\) and the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Find also the envelope of the line \(lx+my=1\), if these two straight lines are at right angles.
A light straight uniform rod of circular section is held horizontal, and is then slightly bent by vertical forces of amount \(w\) per unit length. Prove that the displacement \(y\) of the neutral line at a point distant \(x\) from one end of the rod is given by \[ EI \frac{d^4 y}{dx^4} = w, \] where \(E\) is Young's modulus, and \(I\) the moment of inertia of the cross section about a diameter. Explain how \(y\) is determined if the load is concentrated in various points, and show that the displacement at \(Q\) due to a concentrated load at \(P\) is equal to the displacement at \(P\) due to an equal load at \(Q\), it being supposed that the rod is "supported" at its ends.
A particle is moving on the inside of a rough circular cylinder whose radius is \(a\) and axis vertical. Establish the equations \[ \dot{V}\phi = -g\sin\phi, \] \[ \frac{V\dot{\phi}}{a}+V^2\sin^2\phi=g\cos\phi, \] where \(V\) is the velocity of the particle at any time, \(\phi\) the angle the direction of motion makes with the downward vertical, and \(\mu\) the coefficient of friction. Find \(V\) and \(\phi\) in terms of \(\dot{\phi}\).
A uniform hollow circular cylinder is free to turn about its axis which is horizontal. A uniform sphere is placed on the top of the cylinder and is slightly disturbed in such a way that its centre moves in a plane perpendicular to the axis of the cylinder in the motion that ensues. Show that slipping will in all cases take place before the sphere leaves contact with the cylinder, and that it commences when \[ 2M\sin\theta = \mu[(17M+6m)\cos\theta - (10M+4m)]. \] \(M,m\) are the masses of the cylinder and sphere, respectively, \(\mu\) is the coefficient of friction, and \(\theta\) the angle a perpendicular from the point of contact on the axis of the cylinder makes with the vertical. It is assumed that the radius of gyration of the cylinder is equal to its radius. What may be deduced from the above equation by writing (i) \(M=0\), (ii) \(m=0\)?
Establish Lagrange's equations of motion for a dynamical system with \(n\) degrees of freedom, where \((n+m)\) generalised coordinates are used in specifying the configuration. A uniform rod has at its ends two smooth rings which slide on a wire in the form of the parabola \[ x^2=4ay, \] the \(y\) axis being vertical and upwards. Give equations of motion for the general case, and show that the horizontal position of equilibrium is stable if \(l<2a\), \(2l\) being the length of the rod.
A body is moving, under gravity, in contact with a smooth horizontal plane. Taking as axes of reference the principal axes at the centre of gravity, write down equations sufficient to determine the motion. A uniform prolate spheroid of semi-axes \(a,c,e\), is rotating about its axis of revolution which is vertical with angular velocity \(\omega\) in contact with a smooth horizontal plane. Shew that the motion is stable if \[ \omega^2 > \frac{5g(a^2-c^2)}{ac^4}. \]
The boundary of a gravitating solid of density \(\rho\) is given by \(r=a[1+\epsilon P_n(\cos\theta)]\) \(\epsilon\) being small. Show that the potential at an external point is approximately \[ -\frac{4\pi a^3\rho}{3}\left[\frac{1}{r}+\frac{3a^n\epsilon P_n(\cos\theta)}{(2n+1)r^{n+1}}\right]. \] If the solid is completely covered by liquid of density \(\sigma\) and of total volume \(\frac{4\pi}{3}(b^3-a^3)\), find the equation of the free surface of the liquid.
An infinite cylinder of any cross section is translated at right angles to its length in liquid at rest at infinity, its velocity, \(U\), being constant and parallel to the \(x\) axis. If \(u,v\) are the components along the axes of the velocity of any point of the liquid, show that \[ p/\rho + \frac{1}{2}\{(u-U)^2+v^2\} = C, \] where \(C\) is a constant. It is supposed that the motion is irrotational, and that no body forces act on the liquid. Express as a line integral the force exerted by the liquid on the cylinder at right angles to its direction of motion, and show that the magnitude of this force per unit length of the cylinder is \(\kappa\rho U\). \(\kappa\) is the circulation defined by \[ \kappa = \int(lv-mu)ds, \] where the integral is taken along any closed path surrounding the cylinder and in a plane perpendicular to its axis, and \((l,m)\) are the direction cosines of the normal to the path.
A mass of liquid is moving irrotationally between two surfaces \(S_1\) and \(S_2\) of which one completely surrounds the other. Show that the kinetic energy of the liquid is \[ -\frac{1}{2}\rho\left[\iint_{S_1} \phi \frac{\partial\phi}{\partial n}dS_1 + \iint_{S_2} \phi \frac{\partial\phi}{\partial n}dS_2\right], \] where \(\phi\) is the velocity potential, \(\rho\) the density, and \(\dfrac{\partial}{\partial n}\) denotes differentiation along a normal drawn into the liquid. The motion is supposed acyclic. A sphere of radius \(a\) is moving with velocity \(U\) in a liquid whose only boundary is an infinite rigid plane. When the sphere is at great distance \(d\) from, and moving perpendicularly to, the boundary, show that the energy of the liquid is approximately \[ \frac{\pi\rho a^3 U^2}{3}\left(1+\frac{3a^3}{8d^3}\right), \] neglecting the fourth and higher powers of \(a/d\).
Long waves are travelling along a straight shallow canal of uniform section. Show that \(\eta\), the elevation of the free surface above the equilibrium level at a distance \(x\) along the canal, satisfies the equation \[ \frac{\partial^2\eta}{\partial t^2} = \frac{gA}{b}\frac{\partial^2\eta}{\partial x^2}, \] \(A\) being the area of the section of the canal, and \(b\) its breadth. Find the corresponding differential equation if \(A\) and \(b\) are functions of \(x\).