Problems

A pile weighing 3 tons is driven into the ground by the falling of a weight of 1 ton from a height of 6 feet. At each blow it is driven in a distance of 2 inches. The impact is inelastic. Shew that the resistance to the movement of the pile is equal to a weight of 13 tons. If there were a coefficient of elasticity \(\frac{1}{4}\), the resistance being unaltered, shew that after the first impact the pile would come to rest before a second impact, and find by how much the pile would be driven in at the first impact.

The total mass of a train is 384 tons and the maximum tractive force exerted by the engine at its wheels is 12 tons wt. This force is exerted until the speed of the train from rest reaches 20 feet per second, after which the engine exerts a constant horse power at its wheels. The resistance of the train at various speeds is given by the following table: \begin{center{tabular}{lcccccc} Speed in ft. per sec. & 0 & 5 & 20 & 40 & 60 & 80 \\ Resistance in tons wt. & 2.5 & 0.8 & 1 & 1.4 & 2 & 2.6 \end{{tabular} Shew by means of a curve the acceleration of the train at various speeds. Estimate the time taken to reach a speed of 80 feet per second.

A particle hangs by an inelastic string of length \(a\) from a fixed point, and a second particle of the same mass hangs from the first by an equal string. The whole moves with uniform angular velocity \(\omega\) about the vertical through the point of suspension, the strings making constant angles \(\alpha\) and \(\beta\) with the vertical. Shew that \[ \tan\alpha = \frac{a\omega^2}{g}(\sin\alpha + \frac{1}{2}\sin\beta), \] and \[ \tan\beta = \frac{a\omega^2}{g}(\sin\alpha + \sin\beta). \] Hence shew that if \(\alpha\) and \(\beta\) are small such a steady motion is only possible if \(a\omega^2\) has one of the values \[ (2\pm\sqrt{2})g \] and that \(\beta/\alpha = \pm\sqrt{2}\).

Two particles of masses \(m\) and \(m'\) travelling in the same straight line collide. Shew that the impulse \(I\) between them is given by \[ I\left(\frac{1}{m}+\frac{1}{m'}\right) = U+U', \] where \(U\) is the relative velocity of approach before the impulse and \(U'\) the relative velocity with which they separate. Shew also that the loss of kinetic energy is \(\frac{1}{2}I(U-U')\), and express this in terms of the initial motion and the coefficient of elasticity. Two pendulums of equal length have small spherical bobs of masses \(m\) and \(m'\) which hang in contact with one another. The bob \(m\) is drawn aside through an angle \(\theta\) and allowed to fall so as to strike the second, which comes to rest after turning through an angle \(\theta'\). Shew that the coefficient of elasticity is \(\frac{(m+m')\sin\frac{1}{2}\theta'}{m\sin\frac{1}{2}\theta}-1\).

A flywheel of moment of inertia \(I\) is set in motion from rest by a constant couple \(G\), there being a frictional resistance equal to \(\mu\omega^2\), where \(\omega\) is the angular velocity. Shew that the time taken to attain the angular velocity \(\omega\) is \[ \frac{I\omega_0}{G}\tanh^{-1}\frac{\omega}{\omega_0}, \] where \(\omega_0\) is the limiting velocity \((G/\mu)^{\frac{1}{2}}\). Shew also that the angle turned through from rest is \[ -\frac{I}{2\mu}\log\left(1-\frac{\omega^2}{\omega_0^2}\right). \]

A sequence of terms \(u_1, u_2, \dots, u_n, \dots\) is such that any four consecutive terms are connected by the relation \[ u_n - 4u_{n-1} + 5u_{n-2} - 2u_{n-3} = 0. \] If \(u_1=1, u_2=0, u_3=-5\), find \(u_n\).

Find the relation between \(p\) and \(q\) necessary in order that the equation \(x^3-px+q=0\) may be put into the form \[ (x^2+mx+n)^2 = x^4. \] Hence or otherwise solve the equation \[ 8x^3 - 36x + 27 = 0. \]

Prove that

1. [(i)] \(\displaystyle\frac{1}{4} + \frac{1.3}{4.6} + \frac{1.3.5}{4.6.8} + \frac{1.3.5.7}{4.6.8.10} + \dots \text{ to infinity} = 1\);
2. [(ii)] \(\displaystyle\frac{1}{1.2.3} + \frac{1}{3.4.5} + \frac{1}{5.6.7} + \frac{1}{7.8.9} + \dots \text{ to infinity} = \log_e 2 - \frac{1}{2}\).

\item[(i)] If there are \(n\) straight lines, produced indefinitely, lying in one plane, and no three of them meet in a point, prove that the number of \(n\)-sided polygons they form is \(\frac{1}{2}(n-1)!\). \item[(ii)] Prove that the arithmetic mean of all numbers less than \(n\) and prime to it is \(\frac{1}{2}n\). \item[(i)] Solve the equation \[ \cos x + \cos(x-\alpha) = \cos(x-\beta) + \cos(x-\alpha+\beta). \] \item[(ii)] Prove that \[ \left(1-\tan^2\frac{x}{2}\right)\left(1-\tan^2\frac{x}{2^2}\right)\left(1-\tan^2\frac{x}{2^3}\right)\dots \text{ to infinity} = \frac{x}{\tan x}. \]

A man standing at a distance \(c\) from a straight line of railway sees a train standing on the line, having its nearer end at a distance \(a\) from the point in the railway nearest him. He observes the angle \(\alpha\), which the train subtends, and thence calculates its length. If in observing \(\alpha\) he makes a small error \(\theta\), prove that the percentage error in the calculated length of the train is \[ \frac{100c\theta}{\sin\alpha(c\cos\alpha-a\sin\alpha)}. \]

Prove that the radius \(R\) of the circle that touches externally each of three circles of radii \(a, b, c\), that touch one another externally, is given by \[ \{Rbc(b+c+R)\}^{\frac{1}{2}} + \{Rca(c+a+R)\}^{\frac{1}{2}} + \{Rab(a+b+R)\}^{\frac{1}{2}} = \{abc(a+b+c)\}^{\frac{1}{2}}. \]