An engine, working at the rate of 400 horse-power, is pulling a train, which with the engine weighs 150 tons, up an incline of 1 in 120 at a steady speed of 30 miles an hour. Find the average frictional resistance, expressed in lbs. per ton. The brake-van, weighing 20 tons, becomes detached. How far will it run before stopping? If the engine continues to work at the same horse-power, what will be the value of the speed when it again becomes steady?
The penetration of a 4-ounce bullet at velocity 500 feet per second in a fixed block of wood is 5 inches. If a similar bullet strikes a block of the same wood, 3 inches thick and weighing 1 lb., at the same velocity, shew that if the block is free to move it will be perforated, and find the velocity with which the bullet will issue. It is assumed that the force resisting penetration is constant. State any dynamical laws or principles on which your solution depends.
A particle executes simple harmonic motion in a straight line. Obtain a formula connecting the period, the amplitude, and the maximum velocity. Shew further that the motion is determinate if the velocities \(u, v, w\) at three points \(x=a,b,c\) of the path are known, the origin being anywhere on the path; and that the period \(T\) is given by the equation \[ \frac{4\pi^2}{T^2}(b-c)(c-a)(a-b) = \begin{vmatrix} u^2 & v^2 & w^2 \\ a & b & c \\ 1 & 1 & 1 \end{vmatrix}. \]
Differentiate with respect to \(x\)
Find the conditions that \(f(x)\) should have a minimum value when \(x=a\). An open rectangular tank whose depth is \(y\) and base a square side \(x\) [inside measurements, in feet] is to have an inside capacity \(a^3\) cubic feet. It is made of two pieces of metal, riveted at the four sides of the base and along one of the vertical sides. If the cost of riveting is \pounds \(b\) per foot length of riveted seam measured inside, prove that for the cost of riveting to be a minimum the depth of the tank should be four times its width.
Prove that the length of the subnormal of the curve \[ y=f(x) \text{ is } y\frac{dy}{dx}. \] In the catenary \[ y=c\cosh\frac{x}{c}; \] prove that the subtangent is \(c\coth\frac{x}{c}\), the subnormal is \(\frac{1}{2}c\sinh\frac{2x}{c}\), and the normal is \(\frac{y^2}{c}\).
If \(y=a+x\sin y\), where \(a\) is a constant, prove that, when \(x=0\), \[ \frac{dy}{dx} = \sin a, \text{ and } \frac{d^2y}{dx^2} = \sin 2a. \] Hence by Maclaurin's Theorem expand \(y\) in powers of \(x\) as far as \(x^2\).
If forces are represented in magnitude and direction by \(\lambda \cdot OA, \mu \cdot OB, \nu \cdot OC, \dots\), prove that their resultant is represented in magnitude and direction by \((\lambda+\mu+\nu+\dots)OG\), where \(G\) is the centre of gravity of masses \(\lambda, \mu, \nu, \dots\) at \(A, B, C, \dots\). \(P, Q, R\) are taken on the sides \(BC, CA, AB\) of a triangle dividing each in the same ratio \(1+\lambda:1-\lambda\) in the same sense round the triangle. Prove that forces represented by \(AP, BQ, CR\) are equivalent to a couple whose moment is \(2\lambda\Delta\), where \(\Delta\) is the area of the triangle.
A rectangle is hung from a smooth peg by a string of length \(2a\) whose ends are fastened to two points on the upper edge at distances \(c\) from the middle point. Show that an oblique position of equilibrium is possible if the depth of the rectangle is less than \(\displaystyle\frac{2c^2}{\sqrt{a^2-c^2}}\).
\(ABCD\) is a rhombus of freely jointed rods in a vertical plane and \(B, D\) are connected by a rod jointed to the rhombus. \(A\) and \(B\) are fixed so that \(AB\) is horizontal and below the level of \(CD\). The acute angle \(A\) of the rhombus is \(\alpha\). If a weight \(W\) is hung from \(C\), draw the force diagram and find the stress in the rod \(BD\) in terms of \(W, \alpha\).