Problems

Write down the most general values of \(x\) which satisfy the equations

1. [(i)] \(\sin x = \sin \alpha\),
2. [(ii)] \(\cos x = \cos \alpha\),
3. [(iii)] \(\tan x = \tan \alpha\).

Find all sets of values of \(x, y, z\) which satisfy the simultaneous equations \begin{align*} \tan x + \tan 2y &= 0, \\ \tan y + \tan 2z &= 0, \\ \tan z + \tan 2x &= 0, \end{align*} subject to the restriction that each of \(x, y, z\) is positive and less than \(\pi\).

Prove that the equation of the rectangular hyperbola of closest (i.e. four-point) contact with the parabola \(y^2 = 4ax\) at the point \((a \tan^2\phi, 2a \tan\phi)\) is \[ (x \cos\phi - y \sin\phi + a \sin\phi \tan\phi)^2 = y^2-4ax. \] Prove that in the limit when two of these hyperbolas approach coincidence, they intersect at the point \[ x = -4a - 3a \tan^2\phi, \quad y = 6a \tan\phi + 4a \tan^3\phi. \]

A horizontal portion of a toboggan run is worn into a series of sine-curve undulations 20 ft. from crest to crest, with a maximum height from crest to trough of 1 in. Shew that a toboggan will jump at the crest of an undulation when its speed exceeds about 60 miles an hour.

Define the centre of mass of a system of particles. Prove that, if \(G\) be the centre of mass of a set of \(n\) particles of masses \(m_1, m_2, \dots m_n\) at the points \(A_1, A_2, \dots A_n\), and \(O\) be any point:

1. [(i)] \(\sum_{r=1}^n m_r \vec{OA_r} = \vec{OG} \sum_{r=1}^n m_r,\) where the lengths are added vectorially; and
2. [(ii)] \(\sum_{r=1}^n m_r OA_r^2 = \sum_{r=1}^n m_r GA_r^2 + OG^2 \sum_{r=1}^n m_r.\)

A light equilateral triangular frame is loaded at the corners with weights \(w_1, w_2, w_3\) and suspended from a fixed point by strings of lengths \(l_1, l_2, l_3\) attached to its corners. Prove that the tensions \(T_1, T_2, T_3\) in the strings are given by \[ \frac{w_1l_1}{T_1} = \frac{w_2l_2}{T_2} = \frac{w_3l_3}{T_3} = \frac{\{(w_1+w_2+w_3)(w_1l_1^2+w_2l_2^2+w_3l_3^2) - (w_2w_3+w_3w_1+w_1w_2)a^2\}^{\frac{1}{2}}}{w_1+w_2+w_3}, \] where \(a\) is the length of a side of the triangle.

The internal and external bisectors of the angle \(A\) of the triangle \(ABC\) are drawn meeting \(BC\) in \(D\) and \(D'\). Find the lengths of \(AD\) and \(AD'\) and prove that \[ AD^2+AD'^2 = \frac{4a^2b^2c^2}{(b^2-c^2)^2}. \]

A large number of cards, which are of \(r\) different kinds, are contained in a box from which a man draws one \(n\) times successively. On each occasion it is equally likely that the card drawn will be of any particular kind. Prove that the probability that after drawing \(n\) cards the man will have a complete set, i.e. at least one of every kind, is the sum of the coefficients of all terms in the expansion of \[ (x_1+x_2+x_3+\dots+x_r)^n \] which contain every one of the \(r\) quantities \(x\), divided by \(r^n\).

Two smooth elastic spheres of equal mass are moving in the same direction in parallel paths with velocities \(u, u'\). Prove that, if the spheres impinge, the greatest deviation between their paths after impact is \[ \tan^{-1}\left[\frac{1}{2}(1+e)\left(\frac{u'}{u}-\frac{u}{u'}\right) / \left\{2(1+e^2) + (1-e)^2\left(\frac{u'}{u}+\frac{u}{u'}\right)\right\}\right]. \]

Explain the application of the method of the "funicular polygon" to determine the resultant of a system of coplanar forces. Shew that as applied to a system of vertical coplanar forces \(P, Q, R, \dots\) the method may be employed to construct the bending moment diagram for a straight horizontal beam supported at two given points and loaded by \(P, Q, R, \dots\). Establish the relations which obtain between transverse loading, shear and bending moment in a beam which is subjected to distributed transverse loads. Shew that, if the beam is supported at its ends, the bending moment diagram is parabolic when the loading is uniformly distributed. What is the form of the bending moment diagram when it is similar to the diagram of transverse loading?

Find the equation of the normal at the point \(P(am^2, 2am)\) of the parabola \(y^2 = 4ax\) and the co-ordinates of the point \(Q\) in which it meets the curve again. If \(\phi\) is the angle between the tangents to the parabola at \(P\) and \(Q\) shew that \[ \tan \phi = 2 \tan\theta, \] where \(\theta\) is the angle which the tangent at \(P\) makes with the axis of the parabola.

Prove that, if \(x\) and \(y\) are real, \[ |\cot(x+iy)| < |\coth y|, \quad |\tan(x+iy)| < |\coth y|, \] where \(|a+ib|\) denotes as usual \(+\sqrt{(a^2+b^2)}\).