Six uniform straight rods, each of length \(l\) and weight \(W\), are freely jointed at their ends so that they form the edges of a regular tetrahedron. This tetrahedral framework is suspended by a string attached to one vertex. Calculate the horizontal thrust or tension in each of the horizontal rods. Assuming also a linear dependence of stress on vertical displacement in the sloping rods, calculate the longitudinal thrust or tension in one of these rods at a distance \(x (< l)\) from the highest vertex.
A rocket is propelled vertically upwards by the backward ejection of matter at a uniform rate and with constant speed \(V\) relative to the rocket. The total mass of propelling matter available is \(m\) and it is completely ejected at a time \(\tau\) after launching, when the mass remaining to the rocket is \(km\). Show that, when the time \(t\) is less than \(\tau\), the velocity of the rocket varies according to the equation
\[ \frac{dv}{dt} = -g + \frac{V}{(k+1)\tau - t}. \]
If \(g\tau(k+1)
A uniform straight rod \(AB\) of length \(2a\) and mass \(M\) has a particle of mass \(m\) attached at the end \(B\). The rod is moving freely on a smooth horizontal table when it strikes an inelastic peg fixed in the table. Given that the impact reduces the rod to rest, and that just before the impact \(A\) was instantaneously at rest whilst \(B\) was moving with velocity \(V\), find the position on the rod of the point of impact.
Two particles \(A\) and \(B\) each of mass \(m\) are attached to the ends of a light inextensible string of length \(l\). Initially \(A\) and \(B\) are on a smooth horizontal table, so that the string is taut and at right angles to the edge of the table when \(B\) is pushed gently over the edge. Find (i) the time for \(A\) to reach the edge of the table; (ii) the impulsive tension in the string when \(A\) also leaves the table; (iii) the time between \(A\) leaving the table and \(A\) and \(B\) being next at the same horizontal level; (iv) the horizontal distance moved by \(B\) during the time found in (iii); (v) the vertical distance fallen by \(A\) during the time found in (iii).
Explain clearly and concisely how and why a boy seated on a swing is able to increase the amplitude of successive ``swings'' by his own efforts. Any physical principles assumed in your explanation should be stated precisely both in words and in mathematical terms.
Solve for \(x, y, z\) the three simultaneous equations \[ \begin{cases} ax+3y+2z=b, \\ 5x+4y+3z=1, \\ x+2y+z=b^2, \end{cases} \] explaining in particular the different cases obtained for \(a=3\) with varying values of \(b\).
The cubic equation \(x^3+px+q=0\) has roots \(\alpha, \beta, \gamma\). Find the cubic equation whose roots are \((\beta-\gamma)^2, (\gamma-\alpha)^2, (\alpha-\beta)^2\). Hence or otherwise deduce the condition for the cubic equation \[ a_0x^3+3a_1x^2+3a_2x+a_3=0 \] to have a pair of equal roots in the form \[ (a_0^2a_3+2a_1^3-3a_0a_1a_2)^2+4(a_0a_2-a_1^2)^3=0. \]
Prove that the total number of ways in which a distinct set of three non-zero positive integers can be chosen so that the sum is a given odd positive integer \(p\) is the integer nearest \(p^2/12\).
Two men, \(A\) and \(B\), play a gambling game by tossing together four apparently similar unbiassed coins. If three or four heads are uppermost \(B\) wins, if three or four tails \(A\) wins, and if two of each neither wins. \(A\) and \(B\) start each with \(N\) counters, and after each single game the winner receives a counter from the loser; the first to collect all \(2N\) counters wins the complete game. \(B\), unperceived by \(A\), has arranged that one of the coins is ``double headed'', the other three coins being normal. Show that as a result of this stratagem \(B\)'s chance of winning any single game is four times that of \(A\). Prove further that if when \(A\) has \(r\) counters his chances of winning the complete game is \(U_r\), then \(U_{r+2}-5U_{r+1}+4U_r=0\) for \(0< r< 2N-2\). Hence, or otherwise, deduce that at the start \(A\)'s chance of winning the complete game is \(1/(4^N+1)\). \subsubsection*{SECTION B}
By means of the substitution \[ (1+e\cos\theta)(1-e\cos\phi)=1-e^2 \quad(e<1) \] transform the integral \[ I_n = \int (1+e\cos\theta)^{-n} d\theta \] to an integral involving \(\phi\). Evaluate in terms of \(\phi\) the integrals \(I_0, I_1,\) and \(I_2\).