A uniform wooden pole, of specific gravity 0.64, is floating on water and one end is lifted out of the water. What fraction of the rod's length remains immersed?
Shew (by induction or otherwise) that if \(n\) and \(k\) are positive integers, then \[ f_{n,k} = x^n - k (x+y)^n + \frac{k(k-1)}{1.2}(x+2y)^n - \frac{k(k-1)(k-2)}{1.2.3}(x+3y)^n + \dots \] contains no term in \(x\) of degree higher than \(n-k\); and deduce that \[ f_{n,n} = (-1)^n n! y^n. \]
A straight uniform rod of weight \(w\) and length \(l\) is laid on a rough horizontal table, the coefficient of friction being \(\mu\). A horizontal force is applied at one end at right angles to the rod, and the force is gradually increased until the rod begins to move. Shew that the rod begins to turn about a point distant \(l/\sqrt{2}\) from that end, and that the magnitude of the force then is \(\mu w (\sqrt{2}-1)\).
Any two points \(P, Q\) are taken on two non-intersecting straight lines, shew that the locus of the middle point of \(PQ\) is a plane.
Write a short essay on complex numbers, starting from the beginning and erecting a series of definitions and theorems sufficient to justify the manipulation of complex numbers in accordance with all the laws of elementary algebra.
A simple engine governor consists of a parallelogram of jointed rods each \(9''\) in length: it rotates about a vertical diameter and carries a pair of balls at the side joints each 5 lbs. in weight, whilst the lowest joint carries a collar of 10 lbs. weight sliding on the vertical axis. Find the limits of speed between which the centres of the two balls will rotate at a radius of \(4\frac{1}{2}''\), if there is a frictional force of 2 lbs. weight tending to prevent sliding at the collar.
Shew that the number of divisors (unity and the number itself included) of the number \[ N = p_1^{a_1} p_2^{a_2} \dots p_n^{a_n}, \] where \(p_1, p_2, \dots p_n\) are different primes, is \[ (a_1+1)(a_2+1)\dots(a_n+1) \equiv d(N). \] If \(f(N)\) is the number of different primes, and \(F(N)\) the total number of primes, including repetitions, in the above expression of \(N\), then \[ 2^{f(N)} \leq d(N) < 2^{F(N)}. \] In what circumstances can the signs of equality occur?
A circular cylinder of weight \(W\) rests on a rough inclined plane, being partly supported by a fine thread, which is wrapped round it and, passing over the top of the cylinder, is attached to a fixed point in the plane (the other end is tied to the cylinder and the whole string is in a vertical plane perpendicular to the axis of the cylinder). The string makes an angle \(\gamma\) with the plane. Shew that, if the plane be tilted, the equilibrium is limiting when the angle of inclination, \(\alpha\), of the plane is given by the equation \(\tan\alpha = \mu (1+\cos\gamma)/(1-\mu\sin\gamma)\), and that the tension of the string is then \(\frac{1}{2} W \sin\alpha \sec^2\frac{1}{2}\gamma\).
Prove the following construction for solving graphically the quadratic equation \(x^2 - px + q = 0\). Describe a circle on a diameter whose extremities are the points \((0, 1)\) and \((p, q)\): then the roots of the equation are the abscissae of the points in which this circle cuts the axis of \(x\).
Assuming the logarithmic series, obtain superior and inferior limits for the remainder after \(n\) terms in the expansions in ascending powers of \(x\) of (i) \(\log_e (1+x)\), (ii) \(\log_e \{1/(1-x)\}\), (iii) \(\log_e \{(1+x)/(1-x)\}\). Prove that if these series are used to calculate \(\log_e(128/125)\) correct to ten places of decimals, six terms must be taken in each of the first two series, while two are sufficient in the third case; and, using the tables provided, obtain in each case the remainders correct to two significant figures.