The weight on a suspension bridge is so arranged that the total load carried by the chains including their own weight is uniformly distributed across the span of the bridge. Shew that the curve of the chain is a parabola. The span of the bridge is 100 feet and the sag at the middle of the chain is 10 feet; if the total load on each chain is 25 tons, find the greatest tension in each chain and the tension at the lowest point.
Shew graphically or otherwise that the equation \(10^{x-1} = 2x\) has only two real roots and by means of tables find each of these correct to three places of decimals.
Discuss the conditions of equilibrium of a system of given coplanar forces. Prove that in all cases a system of coplanar forces can be replaced by two forces, one of which acts through a given point and the other along a given straight line.
A nut of given mass and dimensions falls, from rest, down a screw of very steep pitch, fixed with its axis vertical. Compare its motion with that of a body falling freely, neglecting friction. Find the impulse given by a screw press whose motion is suddenly arrested.
Prove that \(2 \cos 5\theta + 1\) is divisible by \(2 \cos \theta + 1\). Find the quotient and employ the result to shew that \[\sec^2 12^\circ + \sec^2 24^\circ + \sec^2 48^\circ + \sec^2 96^\circ = 96.\]
A smoothly jointed framework of light rods forms a quadrilateral \(ABCD\). The middle points \(P, Q\) of an opposite pair of rods are connected by a string in a state of tension \(T\), and the middle points \(R, S\) of the other pair by a light rod in a state of thrust \(X\); shew by the method of virtual work or otherwise that \(T/PQ = X/RS\).
Prove that if \(\frac{1+x}{(1-x)^2}\) is expanded in ascending powers of \(x\) the sum of all the terms after the \(n\)th term is \(a^n\frac{1+x+2n(1-x)}{(1-x)^2}\).
Parallel forces act at given points; shew that their resultant acts at a point independent of their direction, and shew how to determine this point. Given (non-parallel) forces act in a plane at given points; shew that, if the forces are rotated about their points of action through any angle, their resultant rotates through the same angle about a definite point on its line of action.
Explain in general how to draw the curve showing, on an angle base, the turning moment on the crank shaft of a given engine from given indicator diagrams, assuming the load on the engine constant. The turning moment on an engine running at 120 revs. per min. increases uniformly with reference to the angle from zero to 50,000 lbs. ft., and then decreases uniformly to zero at 180\(^\circ\), and is repeated for the second half of the revolution. Determine approximately the moment of inertia of the fly-wheel to procure that the greatest variation of speed is not more than one per cent. above and below the mean speed; the load being constant.
Find the equation of a line perpendicular to the line \(lx + my + n = 0\) and conjugate to it with respect to the ellipse \(x^2/a^2 + y^2/b^2 = 1\), and shew that the two lines determine, on the major axis of the ellipse, a pair of points harmonically related to the foci.