Prove that the radius of curvature at any point of a curve \(y = f(x)\) is \[ \frac{\left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}}}{\frac{d^2y}{dx^2}}. \] Prove that \(\frac{7}{10}a\) is the length of the least radius of curvature of the curve given by \[ x^2y = a\left(x^2 + \frac{1}{\sqrt{5}}a^2\right). \]
Prove that the curve \[ 2x^2y^2+x^3-y^3-2xy=0 \] has (1) a double-point at the origin, each branch having a point of inflexion there, (2) an inflexion at each of the four points \(x=\pm 1, y=\pm 1\), the tangents being parallel to the axes, (3) no rectilinear asymptotes. Give a sketch of the curve.
Prove that if a weight be hung upon the lower end of a vertical spiral spring, it will oscillate vertically with a periodic time equal to that of a simple pendulum of length equal to the static extension of the spring which the weight produces when at rest.
Evaluate \[ \int \frac{x^2+x+1}{x^2(x+1)^2} dx, \quad \int a^x \sin bx \, dx, \quad \int_0^\pi x \sin^2 x \, dx. \]
Prove by induction or otherwise that \[ \int_0^\pi \{\sin n\theta / \sin\theta\} \, d\theta = 0 \text{ or } \pi \] according as \(n\) is an even or odd positive integer.
A ring of mass \(m\) can slide on a smooth circular wire of radius \(a\) in a horizontal plane. The ring is fastened by an elastic string to a point in the plane of the circle at a distance \(c (> a)\) from its centre. Show that if the ring makes small oscillations about its position of equilibrium the period is \(2\pi \left\{\frac{mla(c-a)}{\lambda c(c-a-l)}\right\}^{\frac{1}{2}}\), where \(\lambda\) is the modulus of elasticity of the string and \(l (
Find the area of a loop of the curve \[ r = 3 \sin 2\theta + 4 \cos 2\theta. \]
\(ABE\) is an isosceles triangle, right angled at \(A\). \(BCDE\) is a square on the opposite side of \(BE\) to \(A\). Forces act along the sides of the pentagon \(ABCDE\) represented in magnitude and direction by \(AB, 2BC, 3CD, 4DE, 5EA\). Find the magnitude and line of action of their resultant.
Masses of 3 lbs., 4 lbs., and 5 lbs. hang by strings through three holes in a horizontal table, the other ends of the strings being knotted together. The holes form the vertices of an equilateral triangle of side 3 inches. Find by construction the distances of the knot from the three holes.
A uniform beam \(AB\) of weight \(W\) rests horizontally on two supports at \(C, D\). Weights \(3W, 2W\) are placed at \(E, F\) respectively, where \(AC=3\) ft; \(EF=DB=2\) ft; \(CE=FD=1\) ft. % Diagram of a beam AB on supports C and D is present. % A -- C -- E -- F -- D -- B % Weights 3W at E and 2W at F point downwards. Find the bending moment at \(H\) the point midway between \(C\) and \(D\). If the point of support at \(D\) is raised 6 inches above the level of \(C\) without the beam or the weights slipping, determine whether the bending moment at \(H\) is increased or decreased.