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.
A train of weight \(M\) lb. moving at \(v\) feet per second on the level is pulled with a force of \(P\) lb. against a resistance of \(R\) lb. Show that in accelerating from \(v_0\) to \(v_1\) feet per second, the distance in feet described by the train is \(\int_{v_0}^{v_1} \frac{M}{g}\frac{v\,dv}{P-R}\). If the resistance \(R=a+bv^2\), find an expression for the distance described when the power \(P\) is shut off and the velocity decreases from \(v_0\) to \(v_1\).
A car takes a banked corner of a racing track at a speed \(V\), the lateral gradient \(\alpha\) being designed to reduce the tendency to side-slip to zero for a lower speed \(U\). Show that the coefficient of friction necessary to prevent side-slip for the greater speed \(V\) must be at least \[ \frac{(V^2-U^2)\sin\alpha\cos\alpha}{V^2\sin^2\alpha+U^2\cos^2\alpha}. \]
Prove that \(v\frac{dv}{ds}\) and \(v^2/\rho\) are the tangential and normal components of the acceleration of a particle moving with velocity \(v\) in a plane curve. A particle moves in the curve \(y=a\log\sec\frac{x}{a}\) in such a way that the tangent to the curve rotates uniformly; prove that the resultant acceleration of the particle varies as the square of the radius of curvature. Find the period of oscillation of a particle which moves in a straight line under the action of a force directed to a fixed point in the line and proportional to the distance from that point. Two equal particles connected by an elastic string which is at its natural length and straight, lie on a smooth table, the string being such that the weight of either particle would produce in it an extension \(a\). Prove that if one particle is projected with velocity \(u\) directly away from the other, each will have travelled a distance \(\pi u \sqrt{\frac{a}{8g}}\) when the string first returns to its natural length.