Two normal distributions have different means of 100 and 110 cm and the same standard deviation of 10 cm. A random sample is to be drawn from one of these distributions on the basis of which we have to decide which distribution is being sampled. We wish to have less than 1\% probability of making an error if the distribution is really the one with mean 100 and less than 5\% probability of error in the other case. What is the smallest possible size of sample?
Rain occurs on average on one day in ten. The weather forecast is 80\% correct on days when it is really going to rain and 90\% correct on days when it is going to be fine. On a particular day the forecast is rain. We have to decide whether to stay at home or whether to go out with or without an umbrella. The costs of these actions depend on whether or not it rains and are given in the following table.
The figure represents a suspension bridge. The links forming each chain are pin-jointed; their weight may be neglected. The vertical rods carrying the roadway are equally spaced and equally tensioned. If they are numbered sequentially from one end of the bridge, and the length of the \(n\)th rod is \(y_n\), show that $$y_{n+1} - 2y_n + y_{n-1} = 2k,$$ where \(k\) is a constant. Verify that \(y_n = kn^2\) is one solution of this equation, and find the general solution. Deduce that the ends of the links of each chain lie on a parabola.
A ship enters a lock. When the gates have been closed the ballast tanks in the ship, which contain water, are pumped out into the lock. Discuss with the aid of Archimedes' principle the resulting rise or fall of the surface of the water in the lock. [It follows from Archimedes' principle that, if a body floats at rest in a liquid, the weight of the 'displaced' liquid (i.e. the liquid that would fill the submerged volume of the body) is equal to the weight of the body.]
A uniform circular disc of mass \(M\) and radius \(a\) is placed on a smooth horizontal table. Find from first principles the moment of inertia about a vertical axis through a point of the disc at a distance \(c\) from the centre. If the disc is set in motion by a tangential impulse applied at a point on the edge, determine the point about which the disc will start to rotate.
Particles of masses \(m_1\) and \(m_2\) move in a plane. Show that their kinetic energy is $$\frac{1}{2}(m_1 + m_2)\bar{v}^2 + \frac{1}{2}\frac{m_1 m_2}{m_1 + m_2}V^2,$$ where \(\bar{v}\) is the speed of the centre of mass and \(V\) is the relative speed. A particle of mass \(m\) collides with another particle of mass \(km\) (\(k < 1\)) which is initially at rest, energy being conserved. Find the greatest angle through which the direction of motion of the first particle can be turned.
A comet of mass \(M\) moves under the gravitational attraction \(\mu M/r^2\) of the Sun. Derive from the equations of motion that the total energy, \(\frac{1}{2}M(\dot{r}^2 + r^2\dot{\theta}^2) - \mu M/r\), and the angular momentum about the Sun, \(Mr^2\dot{\theta}\), are constant. If the total energy is zero and the angular momentum has the value \(Mh\), find the differential equation of the orbit in the form $$F\left(\frac{dr}{d\theta}, r\right) = 0$$ and verify that it has solutions $$1/r = 1 + \cos(\theta - \alpha),$$ where \(l = h^2/\mu\) and \(\alpha\) is an arbitrary constant. Calculate the time taken for \(\theta\) to increase from \(\alpha - \frac{1}{2}\pi\) to \(\alpha + \frac{1}{2}\pi\).
A man of weight \(W\) steadily pulls a sledge of weight \(w\) along level ground by means of a rope (of negligible weight) that passes over his shoulder. The breaking strain in the rope is \(W\), and the relevant coefficients of friction between the man and the ground and between the sledge and the ground are both \(\mu = \tan\lambda\). The rope is inclined to the horizontal at an angle \(\theta\). Calculate the tension in the rope, and prove that the inequalities $$w\sin\lambda \leq W\cos(\theta - \lambda), \quad w\cos(\theta + \lambda) \leq W\cos(\theta - \lambda)$$ must both hold.
(i) If the basic units of mass, length and time are changed in such a way that the measures of these quantities are multiplied by factors \(\mu\), \(\lambda\) and \(\tau\) respectively, what is the effect on the measure of a physical quantity that has dimensions \(M^a L^b T^c\)? Explain why the terms in an equation representing a physical relationship must all have the same dimensions. (ii) Either Show that the transformation of physical measures effected by a change of basic units is an element of a commutative group, explaining the meaning to be given to group multiplication (composition). Or A sailing boat has a critical speed \(V\) that cannot ordinarily be exceeded. Assuming that \(V\) depends only on the density of water \(\rho\), the acceleration due to gravity \((g)\), the length of the boat \((l)\) and the shape of the hull, find for a given shape of hull how \(V\) depends on these parameters.
The motion of a rigid body under given forces is unaffected if the following replacements are made: (a) Any two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) acting at a point \(P\) are replaced by the force \(\mathbf{F}_1 + \mathbf{F}_2\) at \(P\), and vice versa. (b) Any force \(\mathbf{F}\) acting at \(P\) is replaced by a force \(\mathbf{F}\) acting at any other point \(Q\) on the line through \(P\) parallel to \(\mathbf{F}\). If a system of forces can be converted into another by a number of such replacements, the systems are said to be equivalent. Prove (i) that any system of forces lying in a plane is equivalent either to a single force or to a couple (i.e. a pair of forces \(+\mathbf{F}\), \(-\mathbf{F}\)), (ii) that two systems of forces lying in the same plane are equivalent if, and only if, they have the same resultant force and the same moment about any one point.