Distinguish between the time-average and the space-average of a varying force acting on a moving body. The force acting on a body of mass 1 lb., which is initially at rest, varies as the square of the time, and is 10 lb. weight at the end of 10 seconds. Neglect resistances and gravity. Prove that the time-average and the space-average of the force during the first 10 seconds are \(3\frac{1}{3}\) lb. wt. and \(6\frac{2}{3}\) lb. wt. respectively.
Find \(\frac{dy}{dx}\) in the following cases:
The volume of water flowing uniformly per second down a given pipe (not quite full) of uniform circular section and uniform slope is proportional to \(\frac{A^{3/2}}{P^{1/2}}\), where \(A\) is the area and \(P\) is the curved portion of the perimeter of the cross-section of the stream. \(P\) subtends a (re-entrant) angle \(\theta\) at the centre of the section of the pipe. Prove that the volume flowing per second is a maximum when \[ 2\theta + \sin\theta = 3\theta\cos\theta. \]
Find the asymptote of the curve \[ x^3+y^3=3axy. \] Sketch the curve, and by transferring to polar coordinates or otherwise prove that the area of its loop is \(\frac{3}{2}a^2\).
State the principle of virtual work for a dynamical system in equilibrium. A uniform lamina in the shape of an equilateral triangle \(ABC\) of side \(a\) has its vertices connected to the vertices of a fixed horizontal equilateral triangle of side \(b\) by equal strings \(AA', BB', CC'\). A couple of moment \(M\) in a horizontal plane acts on the lamina and holds it turned through an angle \(\theta\) from its undisturbed position. Prove that \(\sin\theta = \frac{3hM}{abW}\) where \(W\) is the weight of the lamina and \(h\) the distance between the planes \(ABC, A'B'C'\) (in the disturbed position).
The figure shows a plate gripped by two cylinders which lean against it, the cylinders being hinged at \(A, B\) to fixed supports. The coefficient of friction between each cylinder and the plate is \(\mu\). The masses are \(m, M\) as shown. Show that the plate will not slip if \[ \frac{M}{m} < \frac{2\mu\cos\alpha}{\sin\alpha - \mu(1+\cos\alpha)}. \]
Find the position of the centre of gravity of a uniform semicircular disc. If any point \(P\) is taken upon the diameter \(AB\) of such a disc and the semicircles upon \(AP, BP\) as diameters are removed, find the position of centre of gravity of the part remaining. Show that for different positions of \(P\) the centre of gravity lies midway between \(P\) and a certain fixed point.
Show that a cylinder resting on a rough horizontal plane is in stable equilibrium if the centre of gravity is below the centre of curvature at the point of contact. Show that if the cylinder is elliptic and uniform, the stability of its equilibrium can be disturbed by attaching a heavy particle to it at the highest point, provided the eccentricity does not exceed a certain value.
A rod moves in any manner in a plane; show that it may at any instant be considered to be turning about a point \(I\) (instantaneous centre) in that plane. A circle and a tangent to it are given. A rod moves so that it touches the circle and one end is upon the tangent. Show that the loci of \(I\) in space and relative to the rod are both parabolas.
Show that if a smooth sphere of mass \(m_1\) collides with another smooth sphere of mass \(m_2\) at rest, and is deflected through an angle \(\theta\) from its former path, the sphere of mass \(m_2\) being set in motion in a direction \(\phi\) with the former path of \(m_1\), then \(\tan\theta = \frac{m_2\sin 2\phi}{m_1-m_2\cos 2\phi}\), both spheres being perfectly elastic.