A rigid roof-frame \(ABC\) is in the form of an isosceles triangle with a right angle at \(B\), and rests upon two walls at \(A, C\). It carries a weight \(W\) symmetrically distributed on the two sides, and due to wind pressure there is a force \(w\) uniformly spread over \(BC\) and perpendicular to it. If the reaction at \(C\) is vertical, find the horizontal and vertical components of the reaction at \(A\).
The figure shows a uniform log of square section split along a plane \(EF\) parallel to \(BC\) and resting in equilibrium upon two smooth horizontal parallel rails at the same level so that \(AC\) is vertical. Show that the coefficient of friction between the two faces \(EF\) must at least be \(BE/EA\).
A parallelogram of light rods, smoothly jointed, is put in a state of stress by two strings, each having its ends attached to points on a pair of opposite sides. Show that the tensions of the strings are proportional each to the length of the intercept made on the line of the string by the sides which it does not connect.
Show that in a uniform chain at rest under gravity, the tension at any point is proportional to the height of the point above a certain level. A uniform chain of length \(l\) has its end link free to slide on a smooth vertical wire and passes over a smooth peg at a distance \(a\) from the wire. The other end is attached to a weight equal to \(n\) times the weight of a length \(a\) of the string. Show that for equilibrium to be possible, \(n+l/a\) must not be less than \(e\) (the base of logarithms).
Two small spheres \(A, B\) of equal mass \(m\), are suspended in contact by two equal vertical strings so that the line of centres is horizontal. The sphere \(A\) is drawn aside through a small distance and let go, its velocity on impact being \(u\). Show that all the subsequent collisions occur when the spheres are in the same position as at the first collision, and that the velocity of \(A\) just after the third collision is \(\frac{1}{8}u(1-e^3)\). Show that the kinetic energy of the system tends to the value \(\frac{1}{4}mu^2\).
A string of length \(2l\) has its ends attached to two fixed points \(A, B\), where \(AB=l\), and \(A\) is vertically above \(B\). A bead \(C\) of mass \(m\) can slide freely on the string and describes a horizontal circle with angular velocity \(\omega\) about \(AB\). If \(y\) is the depth of the plane of the circle below \(A\), show that \(y = \frac{l}{2}\left(1 + \frac{4g}{3l\omega^2}\right)\), and find the tension of the string in terms of \(\omega\).
Four equal particles at the corners of a square are connected by light strings forming the sides of the square. If one particle receives a blow \(P\) along a diagonal outwards, show that its initial velocity is \(P/2m\), and find the initial velocities of the other particles.
A gun of mass \(M\) fires a shell of mass \(m\); the elevation of the gun is \(\alpha\) and there is a smooth horizontal recoil. Show that the horizontal range \(x\) due to the development of energy \(E\) in the process of firing is given by \[ xmg(1+m\cos^2\alpha/M) = 4E\sin\alpha\cos\alpha. \] Show also that the maximum range is obtained when \(\tan^2\alpha = 1+m/M\), and is equal to \[ 2E/(mg(1+m/M)^{\frac{1}{2}}). \]
A uniform solid circular cylinder makes complete revolutions under gravity about a horizontal generator. Show that the supports must be able to bear at least \(11/3\) times the weight of the cylinder.
Points \(P, Q\) are taken in the sides \(AB, CD\) respectively of a quadrilateral \(ABCD\) so that \(AP:PB::CQ:QD\). The line parallel to \(AB\) through \(D\) meets \(AQ\) at \(R\), and the line parallel to \(PQ\) through \(R\) meets \(BQ\) at \(S\). Prove that \(CS\) is parallel to \(AB\). Hence prove that the area of the quadrilateral is equal to the sum of the areas of the triangles \(AQB, CPD\).