Describe the type of fracture to be expected when round bars of good mild steel and good grey cast iron are broken (a) in tension and (b) in torsion. If the bars are 1" in diameter, suggest suitable values of the breaking load for each material in the two cases. Distinguish between "hardening" and "tempering" of high carbon steel. How is mild steel case-hardened?
The moment of inertia of cross-section of a cantilever of length \(l\) varies from \(I\) at the support to \(\frac{1}{4}I\) at the outer end, where a load \(W\) is carried. Shew that the deflection at the outer end is \(\frac{3Wl^3}{8EI}\log_e 3\).
The rotating parts of a motor-car engine may be considered as equivalent to a flywheel weighing 100 lb. and having a radius of gyration of 4\(\frac{1}{2}\)". When the engine is running at 1000 r.p.m. the clutch is accidentally let in suddenly with the car at rest and the gear engaged. If the clutch slips for \(\frac{1}{10}\) sec., after which the engine speed is 200 r.p.m., find the maximum stress reached in the clutch-shaft, which is tubular, with external and internal diameters of 1" and \(\frac{3}{4}\)" respectively.
On a thick cylinder, whose external and internal diameters are 6" and 4" respectively, is wound one layer of steel wire under a uniform tension during winding of 1000 lb., the diameter of the wire being 0.1". Calculate the greatest intensity of "hoop" compressive stress in the tube.
Obtain a geometrical construction for dividing a line into two parts so that the rectangle contained by the whole line and one part may be equal to the square on the other part. Shew how this mode of division may be used to construct a regular pentagon. Obtain a construction for dividing a line so that the square on one part may be \(n\) times the rectangle contained by the whole line and the other part, where \(n\) is any integer.
If \(P\) is a point in the plane of the triangle \(ABC\) and \(\alpha.PA^2 + \beta.PB^2 + \gamma.PC^2 = \delta\), where \(\alpha, \beta, \gamma, \delta\) are constants, prove that the locus of \(P\) is a circle. Hence, or otherwise, find the position of the point \(P\) when \(PB^2+PC^2-PA^2\) is a minimum; and shew that \(CA.PB^2+AB.PC^2-BC.PA^2\) is a minimum at the centre of one of the escribed circles of the triangle \(ABC\). State the minimum value in each case.
Prove Pascal's theorem, and shew by means of it how to construct any number of points on a conic
Shew how to find the focus, directrix and eccentricity of the section of a circular cone by any plane. If a circular cone of vertical angle \(2\alpha\) is cut by a plane so that the greatest and least distances of the curve of section from the vertex are \(r, r'\), shew that the minor semi-axis of the section is \((rr')^{1/2}\sin\alpha\). Hence shew that the envelope of planes which cut a given circular cone in sections having a minor axis of given length is a hyperboloid of revolution.
Shew how to draw a line through a given point to meet two given non-intersecting lines. If \(A, B, C\) are three non-intersecting lines shew that there is an infinity of lines \(L_r\) meeting all three. If the plane of \(L_r\) and \(A\) meets a fourth line \(D\) in \(P_r\), and the plane of \(L_r\) and \(B\) meets \(D\) in \(Q_r\), shew that there is in general a one-one correspondence between \(P_r\) and \(Q_r\) and hence that \(D\) meets either two or all of the lines \(L_r\).
Obtain the equation of the pair of tangents from a point \((x_1, y_1)\) to the ellipse \(\displaystyle\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). A point \(P\) moves so that the part of a fixed tangent to the ellipse intercepted between the tangents from \(P\) to the ellipse subtends a right angle at the centre. Shew that the locus of \(P\) is a straight line touching the ellipse \(\displaystyle\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{a^2+b^2}{a^2-b^2}\).