Find formulae of reduction for \[ \int (1+x^2)^n dx, \quad \int e^x \sin^n x dx. \]
Shew that the area bounded by the parabola \(ay=x^2\) and the lines \(y=x, y=2x\) is \(\frac{7}{6}a^2\).
State the laws of (i) limiting friction, and (ii) rolling friction. A uniform rod \(AB\) of weight \(W\) and length \(l\) lies on a horizontal plane whose coefficient of friction is \(\mu\). A string is attached to \(B\) and is pulled in a horizontal direction perpendicular to the rod. As the tension is gradually increased shew that the rod begins to turn about a point in it whose distance from \(A\) is approximately \(\frac{3l}{16}\), and the tension of the string is then about \(\frac{2\mu W}{5}\).
A uniform rod, of length \(c\), rests with one end on a smooth elliptic arc whose major axis is horizontal and with the other on a smooth vertical plane at a distance \(h\) from the centre of the ellipse; the ellipse and the rod both being in a vertical plane. Prove that, if \(\theta\) is the angle which the rod makes with the horizontal, and \(2a, 2b\) are the axes of the ellipse, \[ 2b\tan\theta = a\tan\phi, \] where \[ a\cos\phi+h=c\cos\theta; \] and explain the result when \[ a=2b=c, h=0. \]
A regular pentagon \(ABCDE\), formed of light rods, jointed at the angles, is stiffened by two light jointed bars \(AC, AD\). Two equal and opposite forces, each equal to 3 lbs. weight, are applied at \(B\) and \(E\): find graphically or otherwise the stress in each bar of the framework, stating whether it is tensile or compressive.
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.