A commercial process is governed by the equation \(\ddot{x} + 3\dot{x} - 4x = 0\). At the first time \(T\) that \(|x(T)| = 100\) the process must be shut down at once. The total profit made on that run is then \(T\) thousand pounds. Unfortunately the initial values \(x(0)\), \(\dot{x}(0)\) cannot be controlled exactly and all that can be done is to ensure that \(|x(0)|, |\dot{x}(0)| \leq 1\). Estimate the minimum value of \(T\). A machine is available which would improve the accuracy with which \(x(0)\), \(\dot{x}(0)\) are controlled in such a way that \(|x(0)|, |\dot{x}(0)| \leq 1/10\) but this would cost an extra \(S\) thousand pounds per run. Would you advise the use of such a machine if \(S = 1\), if \(S = 4\) or if \(S = 10\) and why? (Clearly you have not got as much information as you might want in ideal circumstances but you do have enough information to come to a sensible decision.) [log\(_e\)10 = 2.30256.]
A road is to be built from a town \(A\) with map coordinates \((x,y) = (-1, -1)\) to a town \(B\) at \((1, 1)\). The cost per unit length of a road in the region \(y \leq 0\) is \(K\) million pounds and that in the region \(y > 0\) is 1 million pounds. The road is to run in a straight line from \(A\) to a point \(C\) at \((u, 0)\) and then in a straight line from \(C\) to \(B\). The total cost will thus be \((K \times \text{length }AC + \text{length }CB)\) million pounds and \(C\) is chosen to minimize this total cost. Let \(\theta\) be the angle between \(AC\) and the negative real axis and \(\phi\) the angle between \(BC\) and the positive real axis. Describe how \(u\) varies with \(K\) (an explicit formula is not required) and give a simple explicit formula for \(\frac{\cos \theta}{\cos \phi}\) in terms of \(K\).
Suppose \(f\) is a twice differentiable function with \(f''(x) < 0\) for all \(x > 0\). Show that if \(0 < a < b\) then \[f(\lambda a + (1-\lambda)b) \geq \lambda f(a) + (1-\lambda)f(b)\] for all \(1 \geq \lambda \geq 0\). By induction or otherwise deduce that if \(a_1, a_2, \ldots, a_n > 0\) then \[f\left(\frac{1}{n}\sum_{j=1}^n a_j\right) \geq \frac{1}{n}\sum_{j=1}^n f(a_j).\] Setting \(f(x) = \log_e x\) deduce that \[\frac{1}{n}\sum_{j=1}^n a_j \geq (a_1 a_2 \ldots a_n)^{1/n}.\] [Hint: Consider \(g(\lambda) = f(\lambda a + (1-\lambda)b) - \lambda f(a) - (1-\lambda)f(b)\) as a function of \(\lambda\).]
Three sets \(A\), \(B\), \(C\) are chosen at random in such a way that: (i) For any one of the sets \(A\), \(B\), \(C\) the event that it has non-empty intersection with either of the other two sets is independent of the event that it has non-empty intersection with the third set, and this event has constant probability. (ii) For any two of \(A\), \(B\), \(C\) the probability that they have non-empty intersection is \(p_1\). (iii) The probability that each pair of \(A\), \(B\), \(C\) has non-empty intersection is \(p_2\). Show that the probability that no two of \(A\), \(B\), \(C\) have non-empty intersection is \[p_0 = 1 - 3p_1(1-p_1) - p_2.\]
Analyse the following cases by any methods you think suitable, explaining briefly in each case the principles underlying your method.
\(P\) is a passenger on a roundabout at a fair. When the roundabout is rotating uniformly, a given point \(A\) on the roundabout moves in a circle of radius \(3a\) about the central axis of the roundabout with constant angular velocity \(\omega\). The passenger \(P\) is moving relative to the roundabout in a circle of radius \(a\) about \(A\) with constant angular velocity \(2\omega\). How many times is \(P\) stationary during one revolution of \(A\)? Find the distance travelled by \(P\) between two points when he is at rest.
A square-wheeled bicycle is ridden at constant horizontal speed \(V\). The sides of one wheel are always parallel to the sides of the other, and the wheels do not slip on the ground. If the wheels remain in contact with the ground show that \[V^2 < ag/16,\] where \(a\) is the circumference of each wheel.
Identical ball-bearings \(A\), \(B\), \(C\), of diameter \(a\), are collinear. \(B\) and \(C\) are initially at rest with their centres a distance \(b\) apart, and \(A\), moving co-linearly towards them, strikes \(B\). The coefficient of restitution is \(e\). Show that \(A\) will strike \(B\) again, after \(A\) has travelled a further distance \[\frac{1+e}{1-e}(b-a).\]
Suppose that the coefficient of friction between two surfaces is directly proportional to the velocity difference between the surfaces (the 'slipping velocity'). Show that a body can slide down an inclined plane with a constant velocity which depends on the inclination of the plane. A cylinder of radius \(a\) and radius of gyration \(k\), with its axis horizontal, rolls (with slipping of the above character) down an inclined plane. If the cylinder starts from rest, determine the subsequent motion. Show that the frictional force tends to the force which would be necessary to keep the cylinder rolling without slipping, but that the slipping velocity at the point of contact does not tend to zero.
A particle of unit mass moves, in the absence of gravity, in the plane of a disc of unit radius and moment of inertia \(k^2\). The particle is attached by a light inextensible string of length \(l_0\) to a point on the rim of the disc. The particle's motion is such that the string is always taut, and wraps itself round the disc as the particle moves in a clockwise sense. (i) If the disc is fixed, show that the kinetic energy \(T\) of the system is \(\frac{1}{2}l^2\dot{\theta}^2\), and its angular momentum \(h\) is \(l^2\dot{\theta}\), where \(l\) is the length of the unwrapped portion of the string. Which, if either, of \(T\) and \(h\) is constant? (ii) If the disc is free to rotate about its axis, with angular velocity \(\dot{\phi}\) which is positive if the body rotates anticlockwise, show that \[h = l^2\dot{\theta}+(1+k^2+l^2)\dot{\phi},\] and find \(T\). Which, if either, of \(T\) and \(h\) is now constant? Show that \[h^2+l^2\dot{\theta}^2(1+k^2) = 2T(1+k^2+l^2).\]