A hollow circular cylinder of moment of inertia \(I\) about its axis is initially at rest. It is made to spin about its axis by a motor which applies a constant torque \(G\). The motion is opposed by a frictional torque \((G/\omega_0)\omega\), where \(\omega\) is the angular velocity of the cylinder. Find \(\omega\) as a function of time and show that it tends to a limiting value. When the cylinder is rotating at this limiting rate a particle (whose mass is so small that its effect on the motion of the cylinder is negligible) moves on the inner surface of the cylinder, in a plane perpendicular to the axis, with an initial angular velocity \(2\omega_0\) about the axis. The coefficient of friction is \(\mu\). How much time elapses before the particle rotates at the same rate as the cylinder? [The force of gravity may be neglected.]
An earth satellite experiences a gravitational acceleration \(-\gamma r/r^3\), where \(\mathbf{r}\) is its position vector relative to the centre of the earth. Find the period of a satellite moving in a circular orbit of radius \(r_0\). A space vehicle ejected from this satellite acquires initially an additional radial velocity \(v_0\). Obtain the maximum distance from the earth reached by this vehicle. What is the least value of \(v_0\) for which the space vehicle escapes to outer space?
In the electric circuit below, the charge \(Q\) on the capacitor \(C\) is related to the applied electromotive force \(E\) by the differential equation $$L \frac{d^2Q}{dt^2} + 2R \frac{dQ}{dt} + Q/C = E,$$ where \(L > R^2C\), and \(E(t) = E_0 \cos \omega t\), where \(E_0\) and \(\omega\) are constants. Show that the current \(I(t) (= dQ/dt)\) is ultimately in phase with \(E(t)\) if and only if \(\omega^2 LC = 1\).
(i) Prove that if \(A_1\) and \(A_2\) are any two events $$P(A_1 \cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2).$$ (ii) State the corresponding result for four events. (iii) Four letters are placed at random in four envelopes. Assuming that one and only one letter is right for each envelope, use the result in (ii) to find the probability that all four letters are placed in the wrong envelope. [\((A_1 \cap A_2)\) means that both \(A_1\) and \(A_2\) occur, and \((A_1 \cup A_2)\) means that at least one of \(A_1\) and \(A_2\) occur.]
Each of four players is dealt 13 cards from a pack of 52 which contains 4 aces. Player \(A\) looks at his hand and winks at his partner, Player \(B\), which is a pre-arranged signal that his hand has at least one ace. Player \(B\) winks back to show that he has at least one ace as well. Player \(C\) looks at his hand and sees that he has just one ace. From Player \(C\)'s point of view what is the probability that his partner, Player \(D\), also has at least one ace if (i) he saw the winks and understood their meaning; (ii) he knows nothing about his opponents' signals?
A process for obtaining a new sequence \(v_0, v_1, \ldots\) from a given sequence \(u_0, u_1, \ldots\) is defined as follows: Write down the sequence \(u_0, u_1, \ldots\) and below it write the sequence of first differences \(u_1 - u_0, u_2 - u_1, \ldots\); below that write the sequence of second differences, and so on. The sequence \(v_0, v_1, \ldots\) is then read off down the left-hand vertical column. So, for example, starting with \(1, 1, 2, 6, 24, \ldots\) we get: \begin{align*} 1 \quad 1 \quad 2 \quad 6 \quad 24 \quad \ldots \\ 0 \quad 1 \quad 4 \quad 18 \quad \ldots \\ 1 \quad 3 \quad 14 \quad \ldots \\ 2 \quad 11 \quad \ldots \\ 9 \quad \ldots \end{align*} and the new sequence is \(1, 0, 1, 2, 9, \ldots\) If \(u_n\) is defined by the recurrence relation $$u_{n+1} = (n+1)u_n, \quad u_0 = 1,$$ prove that \(v_n\) is defined by the recurrence relation $$v_{n+1} = (n+1)v_n + (-1)^{n+1}, \quad v_0 = 1,$$ and that \(v_n/u_n \to e^{-1}\) as \(n \to \infty\).
Find the greatest value of \(2^{\frac{1}{2}}(p+q)^{\frac{1}{2}}(1-s)^{\frac{1}{2}}+(s-p)^{\frac{1}{2}}(s-q)^{\frac{1}{2}}\) in the three-dimensional region \(p, q, s \geq 0, p+q \leq s \leq \frac{1}{2}\).
Evaluate $$\int_0^\infty \int_0^\infty e^{-x} \frac{\sin cx}{x} dx dc,$$ and hence evaluate $$\int_0^\infty \frac{\sin x}{x} dx.$$
In a plane three circles of equal radii are drawn through a point. Prove that the circle through their other three intersections has the same radius.
Find a necessary and sufficient condition for the pair of straight lines $$px^2 + qxy + ry^2 = 0$$ to be perpendicular. A variable chord \(PQ\) of a conic \(S\) subtends a right angle at a fixed point \(O\) in the plane of \(S\). Prove that the locus of the foot of the perpendicular from \(O\) to \(PQ\) is in general a circle. Under what circumstances is the locus a straight line?