In a painting process, small charged paint drops move in an oscillating electric field. As a drop of mass \(m\) moves in the \(x\)-direction through the air it experiences a frictional force \(-k\dot{x}\), as well as the oscillating electric force whose amplitude varies in space, \(F(x)\cos\omega t\). When the electric field is weak, the motion of the paint drop can be obtained by successive approximations. To a zeroth approximation there is no electric field and the drop does not move, i.e. \(x \simeq x_0\), a constant. At the next approximation the drop oscillates a small distance about \(x \simeq x_0\) in the electric field which can be evaluated at \(x_0\) in this approximation. Thus, writing \(x \simeq x_0 + x_1(t)\), the first correction is governed by \begin{align*} m\ddot{x}_1 + k\dot{x}_1 = E(x_0)\cos \omega t. \end{align*} Find the forced response (i.e. particular integral for) \(x_1\) which has a frequency \(\omega\). At the following approximation it is necessary to take account of the small difference between the electric field at \(x = x_0 + x_1\) and \(x = x_0\), i.e. \begin{align*} x_1 \left.\frac{dE}{dx}\right|_{x_0} \cos \omega t. \end{align*} Thus, writing \(x \simeq x_0 + x_1(t) + x_2(t)\), the second correction \(x_2\) is governed by \begin{align*} m\ddot{x}_2 + k\dot{x}_2 = x_1(t) \left.\frac{dE}{dx}\right|_{x_0} \cos \omega t. \end{align*} The forced response (i.e. particular integral for) \(x_2\) consists of an oscillation with frequency \(2\omega\) and a steady velocity. Find just the steady velocity. If the paint drifts with the small steady velocity you have just calculated, where does it go?
A garden water sprinkler consists of a straight arm of length \(2l\) pivoted at its centre. The arm rotates in a horizontal plane at a steady angular velocity \(\Omega\), with a rusty pivot exerting a constant couple \(G\) on the arm. Water enters through the central pivot and leaves horizontally through small nozzles at the ends of the arm set at an angle \(\theta\) to the direction of the arm. In unit time a mass \(Q\) of water is discharged with a kinetic energy \(\frac{1}{2}QU^2\) relative to the ground. By considering the angular momentum imparted to the water in unit time, show that the angular velocity of the discharged water is \(G/l^2Q\) at the nozzles. Show further that \(\Omega\) is given by \begin{align*} \Omega l = \left[U^2 - \frac{G^2}{l^2Q^2}\right]^{\frac{1}{2}} \tan \theta - \frac{G}{lQ}. \end{align*} What is the total power which must be supplied to the sprinkler?
Let \(n\) be a non-negative integer. Show that the number of solutions of \[x + 2y + 3z = 6n\] in non-negative integers \(x\), \(y\) and \(z\) is \(3n^2 + 3n + 1\). Find the corresponding number for the equation \[x + 2y + 3z = 6n + 1.\]
Show that the square of any odd integer is congruent to 1 modulo 8. Let \(R\) be the ring of integers taken modulo 8 and let \(G\) be the group of all \(2 \times 2\) matrices \[\begin{pmatrix} u & x \\ y & v \end{pmatrix}\] with entries in \(R\) such that \(u\) and \(v\) are both odd, \(x\) and \(y\) are both even and \(uv - xy = 1\). Determine the number of elements in \(G\) and the number of elements of order 2 in \(G\). [You need not verify that \(G\) is a group.]
The cubic equation \[x^3 + 3qx + r = 0 \quad (r \neq 0)\] has roots \(\alpha\), \(\beta\) and \(\gamma\). Verify that the sextic equation \[r^2(x^2 + x + 1)^3 + 27q^3x^2(x + 1)^2 = 0\] is satisfied by \(\alpha/\beta\). Comment on this result in relation to the roots of the cubic in the cases (i) \(q = 0\) and (ii) \(4q^3 + r^2 = 0\).
Let \(d_1, d_2, ..., d_k\) be the distinct positive divisors of the positive integer \(n\), including 1 and \(n\). Prove that \[(d_1 d_2 ... d_k)^2 = n^k.\]
The equation of the tangent plane to the real ellipsoid \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\] at the point \((x_1, y_1, z_1)\) is \[\frac{xx_1}{a^2} + \frac{yy_1}{b^2} + \frac{zz_1}{c^2} = 1.\] Prove that the common tangent planes to the three ellipsoids \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,\] \[\frac{x^2}{b^2} + \frac{y^2}{c^2} + \frac{z^2}{a^2} = 1,\] \[\frac{x^2}{c^2} + \frac{y^2}{a^2} + \frac{z^2}{b^2} = 1\] touch a sphere of radius \(\{(a^2 + b^2 + c^2)/3\}^{\frac{1}{2}}\), and that the points of contact of these planes with the ellipsoids lie on a sphere of radius \((a^4 + b^4 + c^4)^{\frac{1}{2}}(a^2 + b^2 + c^2)^{-\frac{1}{2}}\).
Let \(\cal S\) be an infinite set of pairs of points in the plane such that the points in question do not all lie on a circle or on a straight line. Show that the following two conditions on \(\cal S\) are equivalent: (i) there is a fixed point \(P\) and a constant \(k\) such that for all pairs \(\{X, X'\}\) in \(\cal S\), \(P\) lies on the line segment \(XX'\) and \(XP.PX' = k^2\); (ii) the four points of any two pairs in \(\cal S\) lie on a circle or line, in an ordering in which the pairs are interleaved. \(T\) is a transformation of the plane \(\Pi\), with an inverse \(T^{-1}\), such that both \(T\) and \(T^{-1}\) send all circles to circles. Let \(\mathcal{C}(P, k)\) be the set of all circles in \(\Pi\) containing a chord \(XX'\) such that \(P\) lies on the segment \(XX'\) and \(XP.PX' = k^2\). Show that \(T\) maps \(\mathcal{C}(P, k)\) to a set of circles of the form \(\mathcal{C}(\tilde{P}, \tilde{k})\). Show that \(T\) can be extended to a transformation of three-dimensional space containing \(\Pi\), that maps all spheres with centres in \(\Pi\) to spheres with centres in \(\Pi\).
Show that \(e^{-t^2/2} \geq \cos t\) for \(0 \leq t \leq \frac{1}{4}\pi\).
Let \(\omega = e^{\pi i/k}\), where \(k\) is an integer greater than 1. Let \(T_0 = 0\) and \[T_j = \omega + \omega^2 +...+ \omega^j.\] Show that \(T_{2k} = 0\), and sketch the polygon \(T_0 T_1...T_{2k}\) in the Argand diagram. Now let \(S_0 = 0\) and \(S_j = \omega + \omega^2/2 +...+ \omega^j/j\). Express \(S_j\) in terms of \(T_0, ..., T_j\) and show that each of the numbers \(S_0, ..., S_{2k}\) lies within or on the polygon \(T_0 T_1...T_{2k}\).