Shew that the locus of centres of a family of conics through four given points is a conic. Shew also that, in the case of a family of conics having three-point contact at P and passing through a fourth point Q, the locus of centres touches the conics at P, has curvature at P of the opposite sign and of double the magnitude of that of the conics, and has PR as a diameter, where R is the middle point of PQ.
Shew that the function \(\sin x + a \sin 3x\) for values of \(x\) from \(0\) to \(\pi\) has no zeroes except the terminal ones if \(-\frac{1}{9}
Find formulae of reduction for the integrals \[ \int \sin^n x dx, \quad \int e^{-ax}\sin^n x dx, \quad \int \frac{dx}{(x^2+a^2)^n}. \] If \[ f(p,q) = \int_0^{\frac{\pi}{2}} (\cos x)^p \cos qx dx, \] shew that \[ f(p,q) = \frac{p(p-1)}{p^2-q^2}f(p-2,q) = \frac{p}{p+q}f(p-1, q-1). \]
A, B are two equal and equally rough weights lying on a rough table and connected by a string. A string is attached to B and is pulled in a direction making an angle \(\alpha\) with AB produced until motion is about to ensue. Examine the cases where one or both of the weights are on the point of motion, shewing that they arise according as \(\alpha \lessgtr \frac{\pi}{4}\). Shew that in the latter case B is about to move in a direction making an angle \(2\alpha\) with AB produced. Find the force needed in each case.
State the general principle of virtual work and prove that when applied to the case of a single rigid body in two dimensions it leads to three independent necessary and sufficient conditions of equilibrium and deduce these conditions in one of their usual forms. Four equal uniform rods, of length \((a+b)\), are hinged at A, B, C, D, and are suspended by two strings of equal length, so that the diagonal AC is vertical and A is the highest point. The strings are attached to two pegs in the same horizontal line and to two points in AB, AD at distance \(a\) from A. Prove that if the rods rest in the form of a square the inclination of the strings to the horizontal is \(\tan^{-1}(a/b)\).
A particle of mass \(m\) is attached by a string to a point on a fixed circular cylinder of radius \(a\) whose axis is vertical. The particle is projected with velocity \(v\) at right angles to the string along a smooth horizontal plane so that the string winds itself round the cylinder. Shew (i) that the velocity of the particle is constant; (ii) that the tension of the string is inversely proportional to the length which remains straight at any instant; (iii) that if the initial length of the string is \(l\) and the breaking tension is \(T\), the string will break when it has turned through an angle \(\dfrac{l}{a}-\dfrac{mv^2}{aT}\).
Prove that if two particles of masses \(m_1, m_2\) are moving in a plane, their kinetic energy is \[ \frac{1}{2}(m_1+m_2)V^2+\frac{1}{2}\frac{m_1 m_2}{m_1+m_2}v^2, \] where \(V\) is the velocity of their centre of mass, and \(v\) is their relative velocity. A shell of mass \((m_1+m_2)\) is fired with a velocity whose horizontal and vertical components are \(U, V\), and at the highest point in its path, the shell explodes into two fragments \(m_1, m_2\). The explosion produces an additional kinetic energy \(E\), and the fragments separate in a horizontal direction: shew that they strike the ground at a distance apart which is equal to \[ \frac{V}{g}\sqrt{2E\left(\frac{1}{m_1}+\frac{1}{m_2}\right)}. \]
State and prove the relation between the moment of inertia of a rigid body about any axis and its moment of inertia about a parallel axis through its centre of gravity. Find the moment of inertia in lb. ft\(^2\). units of a solid cylinder of length 10 inches and diameter 1 inch about an axis perpendicular to its length passing through its centre of gravity. The cylinder is composed of material weighing 0.27 lb. per cubic inch. Shew that if the cylinder is suspended in a horizontal position by two vertical strings of length 3 feet, one attached to each end, the time of a small oscillation in a horizontal plane is about 1.1 seconds.
The shape of the ground forming the bottom of a shallow tidal estuary is such that the area flooded is proportional to the square of the depth of the water at the entrance. The entrance channel has steep banks which may be taken as vertical. At low water the estuary and channel are just empty and as the tide rises the depth of water increases as a simple harmonic function of the time of which the periodic time is 12 hours. If the maximum height of the tide is 12 feet, the channel is 200 yards wide, and the area flooded at high tide 2 square miles, find the velocity of flow in the channel at half tide.