Problems

\(P\) is a variable point on a plane curve \(\Gamma\), and \(R\) is the centre of curvature of \(\Gamma\) at \(P\). Let \(\Delta\) be the locus of \(Q\), where \(Q\) is the mid-point of \(PR\). Show that if \(\phi\) is the angle between the tangent to \(\Gamma\) at \(P\) and the tangent to \(\Delta\) at \(Q\) then \begin{equation*} \tan\phi = \frac{d\rho}{ds}, \end{equation*} where \(\rho = PR\) and \(s\) is the arc length of \(\Gamma\). Prove that if \(\Gamma\) is defined by the equation \(\rho^2 + s^2 = a^2\), then \(\Delta\) is a straight line.

A triangle \(ABC\) is said to be self-conjugate with respect to a circle if \(A\) is the pole of \(BC\), \(B\) is the pole of \(CA\), and \(C\) is the pole of \(AB\). Show that if the triangle \(ABC\) has an obtuse angle there is just one circle with respect to which it is self-conjugate, but that otherwise there is no such circle.

The points \(O\), \(A\), \(B\), \(C\) are not coplanar, and the position vectors of \(A\), \(B\), \(C\) with respect to \(O\) as origin are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) respectively. If \(\mathbf{p}\) is any vector, show that \begin{equation*} [\mathbf{a}, \mathbf{b}, \mathbf{c}]\mathbf{p} = (\mathbf{a} \cdot \mathbf{p})\mathbf{b} \times \mathbf{c} + (\mathbf{b} \cdot \mathbf{p})\mathbf{c} \times \mathbf{a} + (\mathbf{c} \cdot \mathbf{p})\mathbf{a} \times \mathbf{b}. \end{equation*} \(X\), \(Y\), \(Z\) are such that \(X\) is the centre of the sphere through \(O\), \(A\), \(B\), \(C\); \(Y\) is the centre of a sphere which touches the lines \(OA\), \(OB\), \(OC\); and \(Z\) is the second common point of the spheres through \(O\) with centres \(A\), \(B\) and \(C\). Show that the position vectors of \(X\), \(Y\), \(Z\) are of the form \(\mathbf{x}\), \(\lambda\mathbf{y}\), \(\mu\mathbf{z}\) respectively, where \begin{align*} 2[\mathbf{a}, \mathbf{b}, \mathbf{c}]\mathbf{x} &= |\mathbf{a}|^2 \mathbf{b} \times \mathbf{c} + |\mathbf{b}|^2 \mathbf{c} \times \mathbf{a} + |\mathbf{c}|^2 \mathbf{a} \times \mathbf{b}\\ \mathbf{y} &= |\mathbf{a}| \mathbf{b} \times \mathbf{c} + |\mathbf{b}| \mathbf{c} \times \mathbf{a} + |\mathbf{c}| \mathbf{a} \times \mathbf{b}\\ \mathbf{z} &= \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} + \mathbf{a} \times \mathbf{b} \end{align*} and \begin{equation*} \mu = \frac{2[\mathbf{a}, \mathbf{b}, \mathbf{c}]}{|\mathbf{z}|^2}. \end{equation*}

Let \(f_N(a) = \underset{R}{\textrm{Max}}(x_1 x_2 \ldots x_N)\), where \(a \geq 0\), and \(R\) is the region of values determined by \begin{equation*} x_1 + x_2 + \ldots + x_N = a \end{equation*} and \(x_i \geq 0\) for all \(i\). Show that \begin{equation*} f_N(a) = \underset{0 \leq z \leq a}{\textrm{Max}} \{zf_{N-1}(a-z)\} \end{equation*} \((N > 1)\), with \(f_1(a) = a\). Hence show that \begin{equation*} f_N(a) = \frac{a^N}{N^N}. \end{equation*}

The following figures are the additional hours of sleep gained by the use of a certain drug on ten patients:

1. [(i)] Using a significance test discuss whether these results show convincingly that the drug is an effective sleeping pill.
2. [(ii)] In what circumstances is the test you have used valid?

A shell explodes at a vertical height \(h\) above a plane which is inclined at an angle \(\alpha\) to the horizontal; the initial speed \(v\) of the fragments is the same in all directions. Show that the distance between the highest and lowest points of the plane that can be reached by the fragments is \begin{equation*} \frac{2v^2\sec\alpha}{g}\left(\sec^2\alpha + \frac{2gh}{v^2}\right)^{\frac{1}{2}}. \end{equation*} What is the shape of the area on the plane which is under fire?

1. [(i)] An organ pipe is made of a tube of length \(l\); the passage of a sound wave through the air in the pipe is accompanied by oscillatory motion of the particles in the air. If both the velocity \(c\) and the frequency \(f\) of the sound wave depend (at most) on the pressure \(p\), the density \(\rho\) of the air and \(l\), what is the dependence of each of \(c\) and \(f\) on these variables?
2. [(io)] The volume of fluid per unit time flowing down a straight circular pipe of length \(l\) and radius \(a\) is given by \begin{equation*} \frac{\pi a^4(p_1 - p_2)}{8\mu l}, \tag{1} \end{equation*} where \(p_1\) and \(p_2\) are the pressures at the two ends of the pipe and \(\mu\) is the viscosity of the fluid (which is a constant for a given fluid). When a sphere of radius \(r\) moves slowly through a very large volume of fluid with steady velocity \(U\), the resisting force \(F\) depends only on \(r\), \(U\) and \(\mu\). Find how \(F\) depends on \(r\), \(U\) and \(\mu\), by using (1) to find the dimensions of \(\mu\). Indicate how this result would be modified in each of the following cases:

A solid spherical ball of radius \(a\) rolls on a level floor towards a step of height \(h\) \((h < a)\). Initially the angular velocity of the ball is \(\Omega\). Find the condition that the ball fails to mount the step, assuming that the collision is inelastic and that there is no slipping of the ball on the step. If the ball fails to mount the step, and the subsequent collision with the floor is inelastic, prove that the angular velocity with which the ball finally rolls away from the step is \(\omega\) where \begin{equation*} \frac{\omega}{\Omega} = \left(1 - \frac{5h}{7a}\right)^2. \end{equation*}

1. [(i)] Find a vector \(\mathbf{r}\) for which \begin{equation*} \mathbf{r} \times \mathbf{a} + \mathbf{r} = \mathbf{b} \end{equation*} where \(\mathbf{a}\) and \(\mathbf{b}\) are given vectors.
2. [(ii)] If \(\mathbf{u}(t)\) is a vector function of the variable \(t\) and if \(\mathbf{u}\) and \(\dot{\mathbf{u}}\) are unit vectors, show that \begin{equation*} \mathbf{u} \times (\dot{\mathbf{u}} \times \ddot{\mathbf{u}}) = -\dot{\mathbf{u}}. \end{equation*}
3. [(iii)] Evaluate the following expressions, where \(\mathbf{r} = \mathbf{r}(t)\) is a vector function of \(t\), \(r = |\mathbf{r}|\) and dots denote differentiation with respect to \(t\): \begin{equation*} (a) \int \dot{\mathbf{r}} \cdot \ddot{\mathbf{r}} dt, \quad (b) \int \mathbf{r} \times \ddot{\mathbf{r}} dt, \quad (c) \frac{d}{dt}\frac{\mathbf{r}}{r^3}. \end{equation*}

The pendulum of a grandfather clock comprises a thin uniform rod of mass \(m\) and of length \(2a\) which is fixed at one end and a circular disc of mass \(12m\) and radius \(a\) which can be clamped on to the rod so that its centre is on the rod. The clock is designed so that, when the centre of the disc is \(\frac{3}{4}a\) from the fixed end, each half period of the pendulum is exactly one second. By adjusting the position of the disc on the rod a clock can be made to gain or lose time. Neglecting the effect of any mechanism and assuming the pendulum to turn freely about its fixed end in the plane of the rod, calculate the maximum number of minutes which the clock can be adjusted \((a)\) to gain and \((b)\) to lose in one actual hour.