A rough circular cylinder of radius \(r\) is fixed against a smooth vertical wall so that its axis is horizontal. A uniform rectangular plank of length \(2a\) and thickness \(2b\) rests in equilibrium on the cylinder and leaning against the wall with which it makes an angle \(\phi\). The ends of the plank are parallel to the axis of the cylinder and one edge of the lower end is in contact with the cylinder. If \(\lambda\) is the angle of friction between the plank and the cylinder, prove that \[ a\tan\phi < b+2a\tan(\lambda-\theta), \] where \(\theta\) is the acute angle given by \(2a\sin\phi = r(1+\sin\theta)\). If a particle whose weight is equal to that of the plank is now fixed to the upper side of the plank so that the equilibrium becomes limiting, find the distance of the particle from the highest edge of the plank.
Explain the application of Bow's notation in the graphical solution of certain statical problems. The figure represents a framework of freely jointed light rods suspended from \(A\). Equal weights of 100 lb. are suspended from \(G\) and \(E\). Find graphically the stress in each rod, noting which rods are in a state of tension and which in a state of compression. \[ AB=BC=CA, \] \[ BD=DE=EB=CF=FE=EC=2AB, \] \[ DG=GF=3AB. \] \centerline{\includegraphics[width=0.6\textwidth]{framework_diagram.png}} Diagram of a complex framework suspended from A. It has a central triangle ABC. Two larger triangles BDE and CFE are attached. A point G hangs below D and F.
If two smooth rigid bodies moving with given velocities collide, how may their velocities after impact be determined? Consider in detail the case of two smooth spheres and determine the loss of kinetic energy in terms of the velocities before impact and the masses of the spheres. Inside a narrow smooth tube closed at both ends two particles of masses \(M, m\) and moving in opposite directions with velocities \(V, v\) collide. The tube is perfectly elastic and of length \(a\). \(e\) is the coefficient of restitution between the particles. Find the time that elapses between the first and the \(n\)th collisions and also the loss of kinetic energy in the first \(n\) collisions.
\(A\) and \(B\) are two fixed points in the same vertical line and a distance \(a\) apart. A particle of mass \(m\) is attached to \(A\) and \(B\) by two similar light elastic strings each of which has an unstretched length \(l\) and modulus of elasticity \(\lambda\). If \(T\) is the tension in the upper string when the system is in equilibrium, prove that \[ T = \frac{a\lambda+lmg-2l^2\lambda}{2l}. \] Show also that the ratio of the period of the motion when the particle is given a small vertical displacement from its equilibrium position to the period of the motion when the particle is given a small horizontal displacement from its equilibrium position is \[ \left(1-\frac{2al^2\lambda^2}{a^2\lambda^2-l^2m^2g^2}\right)^\frac{1}{2}. \]
Two particles of masses \(M, m\) are connected by a light inextensible string which passes over a smooth peg \(A\). The mass \(m\) is hanging freely but the mass \(M\) lies on a rough plane whose coefficient of friction is \(\mu\). If \(M\) moves down the plane, find its acceleration. The string between \(M\) and \(A\) is parallel to a line of greatest slope of the plane, and the plane is inclined to the horizontal at an angle \(\alpha\). \centerline{\includegraphics[width=0.8\textwidth]{inclined_plane_diagram.png}} Diagram of a mass M on a plane inclined at alpha. A string passes over a peg A to a hanging mass m. M then hits another plane inclined at beta. After moving from rest through a distance \(l\) the mass \(M\) strikes a smooth perfectly elastic plane inclined at an obtuse angle \(\pi-\alpha-\beta\) to the first plane, the line of intersection of the planes being horizontal. At the same instant the string snaps. At what distance from the line of intersection of the planes will \(M\) next strike the second plane?
The acceleration due to gravity, \(g\), at a point on the earth's surface at sea level is given approximately by the formula \(g = g_0(1-a\cos 2\lambda)\), where \(g_0\) and \(a\) are constants and \(\lambda\) is the latitude of the point. At the equator \(g=32.091\) feet per sec. per sec., and at the North Pole \(g=32.252\) feet per sec. per sec. approximately. Discuss the variation of the period of oscillation of a simple pendulum with \(\lambda\), and show that the change in this period due to a small change \(\delta\lambda\) in \(\lambda\) is approximately equal to that due to a percentage shortening of the pendulum of amount \(\frac{\delta\lambda}{2}\sin 2\lambda\).
A function of \(x\) is defined for positive values of \(x\) by the equation \[ f(x) = \int_1^x \frac{du}{u}. \] Without assuming properties of the logarithmic function, prove that for positive values of \(x\), \begin{align*} f(x)+f(y) &= f(xy), \\ f(1/x)+f(x) &= 0, \\ f(\sqrt{x}) &= \frac{1}{2}f(x), \\ \sqrt{1+x^2}-1 &> f(\sqrt{1+x^2}) > \frac{x^2}{2(1+x^2)}. \end{align*} Find limits between which \(f(10)-2f(3)\) must lie.
Show that if \(\phi\) is a function of the coordinates, then \begin{align*} \frac{\partial\phi}{\partial x} &= \cos\theta\frac{\partial\phi}{\partial r} - \frac{\sin\theta}{r}\frac{\partial\phi}{\partial\theta}, \\ \frac{\partial\phi}{\partial y} &= \sin\theta\frac{\partial\phi}{\partial r} + \frac{\cos\theta}{r}\frac{\partial\phi}{\partial\theta}, \end{align*} where \((x,y)\) are rectangular cartesian coordinates and \(x=r\cos\theta, y=r\sin\theta\). Prove that if \(a, b, c\) are constants, then \(\phi=a\log r+b\theta+c\cdot\frac{\sin n\theta}{r^n}\) satisfies the equation \[ \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} = 0. \]
The equation of a plane curve is given in the form \(u=f(\theta)\), where \((1/u, \theta)\) are the polar coordinates of a point. Find an expression for the radius of curvature of the curve in terms of \(u\) and its differential coefficients with respect to \(\theta\). \(S\) is a focus and \(V\) the corresponding vertex of a conic whose latus rectum is of length \(2l\) and whose eccentricity is a small quantity \(e\). Prove that the radius of curvature of the conic at a point \(P\) is \(l(1+\frac{3}{2}e\sin^2\theta-3e^2\sin^2\theta\cos\theta)\), where powers of \(e\) higher than the third are neglected and \(\theta\) denotes the angle \(PSV\). Find also to the same order in \(e\) the distance of the centre of curvature at \(P\) from \(S\).
Explain how to determine the position of the centre of mass of a uniform plane lamina which is bounded by the line \(\theta=0\) and an arc of a curve whose polar equation is given. Find the polar coordinates of the centre of mass of one of the halves of the loop of the curve \(r=a\cos n\theta\) which is bisected by the line \(\theta=0\).