Problems

A uniform disc 16 inches in diameter and 1 inch thick weighs 56 lbs. Small masses of 8, 7, 6 ... 1 oz. are fixed on the upper face at distances of 1, 2, 3 ... 8 inches respectively from the centre at equal angular intervals of 45° round the disc, viz. in the N., N.E., E. ... N.W. directions respectively. If the disc is then suspended from the centre of its upper face, what point of the disc will be lowest and by how much will it be depressed? Graphical methods may be used.

Find graphically, or by methods of approximate integration, the area and the position of the centroid of the deck of a ship, the breadth of the deck being given by the following data, and its length being 320 feet.

Solve completely the equations \[ \sin x = \sin 2y, \quad \sin y = \sin 2z, \quad \sin z = \sin 2x, \] each angle being restricted to be positive and less than \(\pi\).

Shew that \(\cos n\theta\) and \(\frac{\sin n\theta}{\sin\theta}\) are polynomials in \(2 \cos\theta\) of degree \(n\) and \(n-1\) respectively. Prove that

1. [(i)] \(\sin n\theta = 2^{n-1} \prod_0^{n-1} \sin\left(\theta + \frac{r\pi}{n}\right)\);
2. [(ii)] \(n \text{cosec } n\theta = \sum_0^{n-1} (-1)^r \text{cosec } \left(\theta + \frac{r\pi}{n}\right)\), \(n\) odd;
3. [(iii)] \(n \text{cosec } n\theta = \sum_0^{n-1} (-1)^r \cot \left(\theta + \frac{r\pi}{n}\right)\), \(n\) even.

Prove that \((2n+1) \sin \theta - \sin (2n+1)\theta = \sin^3\theta F(\sin^2\theta)\), where \(F\) is a polynomial, and shew that this polynomial never vanishes for real values of \(\theta\).

Express \(\frac{2+x+x^2}{(1+x^2)(1-x)^2}\) as a sum of partial fractions; hence expand the expression in ascending powers of \(x\) (\(x\) small) up to \(x^4\).

Define the terms ``Bending Moment'' and ``Shearing Force.'' Show that if graphs be drawn whose ordinates represent shearing force and bending moment respectively to a base of distance along a loaded beam, any ordinate of the shearing force graph is proportional to the corresponding slope of the bending moment graph. Draw the graphs of bending moment and shearing force for a light horizontal cantilever 10 feet long carrying concentrated loads each of ¼ a ton weight at its centre and at its free end.

A thin smooth rod passes through the centre of a fixed smooth sphere of radius \(a\), projecting beyond it, and is fixed at an angle \(\phi\) to the horizontal. A uniform rod of length \(2l\) rests touching the sphere, and the upper end can slide along the fixed rod by means of a small ring. Show that, if the rod makes an angle \(\theta\) with the vertical in the position of equilibrium in which it is below the fixed rod, \[ a \sin (\phi - \theta) \sin \phi = l \sin \theta \cos^2 (\phi - \theta). \]

Find the necessary and sufficient conditions that, if \(a \neq 0\), \[ ax^2 + 2bx + c \] should be positive or zero for all values of \(x\). Hence or otherwise prove that \[ \{x (b^2 + c^2) + y (c^2 + a^2) + z (a^2 + b^2)\}^2 - 4 (b^2c^2 + c^2a^2 + a^2b^2) (yz + zx + xy) \] is positive for all positive or negative values of \(x, y\) and \(z\), unless \(x/a^2 = y/b^2 = z/c^2\), in which case it is zero.

Explain the geometrical meaning of the expression \(S\), viz., \[ S \equiv x^2 + y^2 + 2gx + 2fy + c, \] where \((x, y)\) are rectangular cartesian coordinates and \(f, g, c\) are constants. Under what conditions does \(S=0\) represent a real locus? Discuss the systems of circles:

1. [(i)] \(b^2x^2 + a^2y^2 - a^2b^2 = (b^2 - a^2)(x - \lambda)^2\),
2. [(ii)] \(b^2x^2 + a^2y^2 - a^2b^2 = (a^2 - b^2)(y - \mu)^2\),

for different values of \(\lambda\) and \(\mu\). Shew that for certain ranges of the values the circles touch the ellipse \(x^2/a^2 + y^2/b^2 = 1\) twice. Determine for what other values of \(\lambda\) and \(\mu\) the circles are real, shewing by a rough figure how they lie. Assume \(a^2 > b^2\).

Prove that, if \(x\) is small compared with \(N^p\), an approximate value of \((N^p + x)^{1/p}\) is \[ N \left ( \frac{2p N^p + (p+1)x}{2p N^p + (p-1)x} \right) \] Show that \(\sqrt{1025}\) is very approximately \(32 \cdot \frac{4099}{4097}\).