Prove that \[ 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n-1}-\log n \] tends to a finite limit, as \(n\to\infty\). If \(p=2^q\), where \(q\) is an integer, prove that \[ \sum_{r=2}^p \frac{1}{r \log r} > \frac{\log(q+1)}{2 \log 2}. \]
Two strings, each of length \(l\), are attached to a ceiling, and the lower ends are attached to a magnet of moment \(M\), length \(l\), and weight \(W\). When the strings are vertical the magnet is in the magnetic meridian but with its north-seeking pole towards the south. Through what angle will it have to turn before it comes to another position of equilibrium? (Assume that the earth's magnetic field exists a couple \(HM \sin\theta\) on the magnet when it makes an angle \(\theta\) with the magnetic meridian.)
The case of a rocket weighs 2 lbs. and the charge 5 lbs. The charge burns at a uniform rate and is completely burnt in 3 seconds, during which time it exerts a constant propulsive force of 20 lb.-wt. If the rocket is fired vertically, find the vertical velocity acquired during the burning of the charge.
A mass is suspended by a light elastic string from a point \(A\) and produces on extension \(k\), the natural length being \(l\). Prove that if the mass is raised through a vertical distance less than \(k\), and is then let go from rest, it makes oscillations of the same period as a simple pendulum of length \(k\). If the mass is raised up to \(A\) and let fall, show that the maximum extension of the string is \(k(1+\sec\alpha)\), where \(\alpha\) is the acute angle given by \(\tan^2\alpha = \frac{2l}{k}\), and that this extension is attained at a time \((\pi-\alpha)\sqrt{\frac{k}{g}}\) after the string first becomes taut.