A simple seismograph consists essentially of a beam \(AB\) free to turn about an axis \(l\) through \(A\) perpendicular to \(AB\), \(l\) being inclined at a small angle \(\phi\) to the vertical and rigidly fastened to the earth. The centre of mass of the beam is at \(G\) on \(AB\) such that \(AG=h\), and the radius of gyration of the beam about an axis through \(G\) parallel to \(l\) is \(k\). When the ground is still, \(AB\) and \(l\) define the neutral plane. At any time \(t\) during an earthquake, the component of the displacement of \(l\) parallel to itself and perpendicular to the neutral plane is \(u\), and the plane through \(l\) containing \(AB\) makes a small angle \(\theta\) with the neutral plane. Neglecting friction, prove that the motion of the beam is given approximately by \[ \frac{d^2\theta}{dt^2} + n^2\theta = -\frac{1}{p}\frac{d^2u}{dt^2}, \] where \(p=(k^2+h^2)/h\) and \(n^2=g\phi/p\). How could the natural period of this instrument be increased? What happens if the period of oscillation of the ground lies near the period of the instrument? Why would the instrument give an unsatisfactory response to a general motion of the ground, and how could its record be made to correspond more closely to local earth-motion (here denoted by \(u\))?
If \(f(n) = \sum_{r=1}^n \csc^2\frac{(2r-1)\pi}{4n}\), prove (by using the identity \(\csc^2\theta + \sec^2\theta = 4\csc^2 2\theta\) or otherwise) that \(f(2n)=4f(n)\), and hence evaluate \(f(2^k)\). Assuming that \(\sin\theta < \theta < \tan\theta\) when \(0 < \theta < \frac{1}{2}\pi\), prove that \[ f(n) > \frac{16n^2}{\pi^2}\sum_{r=1}^n \frac{1}{(2r-1)^2} > f(n)-n. \] Deduce that \[ \lim_{n\to\infty} \sum_{r=1}^{2^n} \frac{1}{(2r-1)^2} = \frac{\pi^2}{8}. \]
Let \(x\) be a real number and let \(f(x)\) denote the fractional part of \(x\), that is \(x-[x]\), where \([x]\) is the largest integer \(\le x\). By considering \(0, f(x), f(2x), \dots, f((n-1)x), 1\) rearranged in increasing order, prove that at least one pair of these numbers differ by at most \(1/n\). Hence prove that we can find integers \(p, q\), where \( 0 < p < n\), such that \[ |px-q| \le 1/n. \]
\(P\) and \(Q\) are two given points on the circumference of a circle, centre \(O\). If a third point \(R\) is taken at random on the circumference of the circle, find in terms of the angle \(POQ\) the probability that the triangle \(PQR\) is acute. Hence show that if three points are taken at random on the circumference of a circle, the probability of their forming an acute-angled triangle is \(\frac{1}{4}\).
Points \(L, M, N\) are taken between vertices on the sides \(BC, CA, AB\) respectively of a triangle \(ABC\). Prove that necessary and sufficient conditions for the circle \(LMN\) to be the incircle of the triangle \(ABC\) are \[ AM=AN, \quad BN=BL, \quad CL=CM. \] \(PQRS\) is a tetrahedron. Prove that necessary and sufficient conditions for the existence of a sphere touching its six edges are \[ QR+PS=RP+QS=PQ+RS. \]
A region \(\mathcal{R}\) of the plane is defined to be convex if for each pair of points \(A, B\), both lying in \(\mathcal{R}\), the whole line segment between \(A\) and \(B\) also lies in \(\mathcal{R}\): \(\mathcal{R}\) is defined to be a polygonal convex region if it is convex and bounded by a polygon. An island forms a polygonal convex region of a flat earth. In times of emergency, civilians are not allowed to be within \(d\) miles of the coast, the value of \(d\) varying with the emergency. Prove that the region to which at any time civilians are restricted is a polygonal convex region. The capital is to be sited so as to lie in all possible civilian regions. Prove that, if \(\mathcal{R}\) is bounded by a quadrilateral \(Q\), this rule determines uniquely the site of the capital, provided \(Q\) is not a trapezium.
Explain the principle of virtual work and discuss its application to problems in statics. Twelve equal uniform rods of weight \(w\) and length \(l\) are freely jointed to form the edges of a regular octahedron. The system is suspended from one corner and a weight \(W\) is attached to the opposite corner. Show that the thrust in a horizontal rod is \[ (W+6w)/2\sqrt{2}. \]
A uniform straight rod of length \(2a\) and mass \(M\) lies on a rough horizontal table with coefficient of friction \(\mu\). A gradually increasing horizontal force is applied to one end in a direction perpendicular to the length of the rod. Show that the rod will move when the force reaches a value \((\sqrt{2}-1)\mu Mg\) and that it begins to turn about a point \(a\sqrt{2}\) from the point of application. (It may be assumed that the vertical reaction is distributed uniformly along the rod.)
A particle is projected with velocity \(v\) and moves freely under gravity. Show that its trajectory is a parabola whose directrix is a horizontal line at a height \(v^2/2g\) above the point of projection. A particle is projected from ground level and passes through two points \(P_1, P_2\) at heights \(h_1, h_2\) respectively above the ground. Show that the velocity of projection is at least \[ \{g(h_1+h_2+d)\}^{\frac{1}{2}}, \] where \(d=P_1P_2\), and also that if the velocity has this minimum value then the focus of the trajectory must lie on the line \(P_1P_2\).
A particle of mass \(4m\) is attached by four elastic strings of natural length \(l\) and elastic modulus \(\lambda\) to the four corners of a horizontal square whose diagonal is \(2a\). Show that the system will be in equilibrium with the strings making an angle \(\theta\) with the horizontal, where \[ a\tan\theta-l\sin\theta = mgl/\lambda. \] Show also that the period of small vertical vibrations is \[ 2\pi\left(\frac{ml}{a}\left(1-\frac{l}{a}\cos^3\theta\right)\right)^{\frac{1}{2}}. \]