A steel pipe of external diameter 3\(\frac{1}{2}\)" and bore 3" carries water at a pressure of 1000 lbs. per sq. inch. If the maximum tensile stress allowed is 4 tons per sq. inch, calculate the greatest allowable eccentricity of the bore.
A shell of mass \(m_1 + m_2\) is projected from a point on a horizontal plane with velocity \(V\) at an angle of projection \(\alpha\) to the horizontal. At the highest point of its path it bursts into two parts of mass \(m_1, m_2\) which separate with a relative velocity which is tangential to the path at the point. If the two parts strike the ground at a distance \(d\) apart, show that the sum of their kinetic energies at impact is greater than the kinetic energy of the undivided shell at the moment of projection by an amount \[ \frac{1}{2}m_1 m_2 g^2 d^2 / \{ (m_1+m_2) V^2 \sin^2\alpha \}. \]
Prove by differentiation (or otherwise) that if \(x>0\), \(\log_e(1+x)\) lies between the sums to \(n\) and \(n+1\) terms of the series \[ x - \frac{x^2}{2} + \frac{x^3}{3} - \dots. \] Deduce that if \(0 < x \le 1\), \(\log_e(1+x)\) is equal to the sum of the series to infinity.
Evaluate the limit as \(x\) tends to infinity of \[ x\{\sqrt{(a^2+x^2)}-x\}. \]
Discuss the properties of simple harmonic motion. Shew that a heavy particle suspended by a light elastic string will oscillate vertically about its position of equilibrium in time independent of the amplitude of oscillation. A particle is suspended from a fixed point by an elastic string of unstretched length \(a\). In the position of equilibrium the extension of the string is \(e_1\); the particle is drawn down a further distance \(e_2\) and set free. Determine the motion completely in the cases when
What is meant by ``Young's Modulus''? Two stiff cross pieces \(A\) and \(A'\) are bolted to the ends of a hollow brass tube \(B\) by steel bolts \(C\) and \(C'\), the dimensions being as shown. [The provided diagram shows two steel cross-pieces, A and A', bolted together by two steel bolts, C and C'. The bolts are positioned outside a hollow brass tube, B, which is placed between the cross-pieces. The length of the tube B is 10 inches. The thickness of each cross-piece is 2 inches. The bolts C and C' are \(\frac{5{8}\) inches in diameter with 14 threads per inch. The brass tube B has an outer diameter of \(1\frac{1}{8}\) inches and an inner diameter of 1 inch.]} The nuts of both bolts are screwed ``hand tight'' in such a manner that the cross pieces are parallel. The nut of \(C\) is then screwed down half a turn with a spanner and the nut of \(C'\) is forced down until the cross pieces are again parallel. Find the forces set up in both bolts and in the tube. The value of Young's Modulus for steel = 14,000 tons per sq. in. The value of Young's Modulus for brass = 5,500 tons per sq. in. The strain in the screw threads and in the cross pieces may be neglected.
A smooth sphere moving with velocity \(V\) on a smooth horizontal plane strikes obliquely in succession two smooth spheres at rest, each equal in all respects to itself, except that the coefficient of elasticity at the first impact is \(e\), and at the second it is unity. If the angle between the line of centres at the first impact and the initial direction of motion of the first sphere is \(\theta\), and the final direction of motion of the first sphere is parallel to its initial direction, show that its final velocity is \[ \frac{1}{2}(1 + \sin^2\theta - e \cos^2\theta)V. \]
Discuss the maxima and minima of \(\tan 3x \cot 2x\), and sketch the general shape of the graph of the function between \(x = \pm \frac{1}{2}\pi\).
Differentiate ab initio \(\log x\), \(\tan^{-1} x\). Differentiate \(e^{\sin(\log x)}\).
Describe the cycle on which (a) a four-stroke gas engine, (b) a Diesel engine, works. Indicate on a diagram in each case the points at which the chief events of the cycle occur, and state briefly the relative advantages of each cycle.