Obtain an expression for the area of a closed oval curve of polar equation \(r=r(\theta)\) in the two cases when the pole (\(r=0\)) is inside the curve and when it is outside the curve. A circle of radius \(a\) has centre \(C\), and a point \(O\) is taken at distance \(c (< a)\) from its centre. The foot of the perpendicular from \(O\) to a tangent to the circle is \(P\). Show that the locus of \(P\) is a closed curve of area \(\pi(a^2+\frac{1}{2}c^2)\).
Two uniform thin rods \(AB, BC\), each of length \(2a\) and of weights \(W_1, W_2\) respectively, are held together rigidly by a bolt and nut at \(B\) so as to form a V, enclosing an angle \(2\beta\); the axis of the bolt is perpendicular to the plane of the V. The rods are hung from a smooth horizontal rail by light rings attached at \(A\) and \(C\). Find the force and the frictional couple exerted at \(B\) by the rod \(BC\) on the rod \(AB\). Show these in a clear diagram. Deduce that the frictional couple at \(B\) may be reduced to zero by the application of equal and opposite horizontal forces at \(A\) and \(C\) of magnitude \(\frac{1}{4}(W_1+W_2)\tan\beta\).
Show that the centre of gravity of a hemispherical bowl, of radius \(a\) and made of uniform thin sheet material, is distant \(\frac{1}{2}a\) from the plane of the rim. The bowl, whose weight is \(W\), can rest in equilibrium with its curved surface in contact with an inclined plane, rough enough to prevent slipping, and a particle of weight \(sW\) fastened to the lowest point of the rim. If \(\beta\) is the inclination of the plane to the horizontal and \(2s = \tan\gamma\), show that \((2\cos\gamma + \sin\gamma)\sin\beta \le 1\).
A uniform rigid square lamina \(ABCD\), of weight \(W\), rests, with the diagonal \(AC\) vertical and \(A\) uppermost, on two parallel horizontal rails which are perpendicular to the plane of the lamina and are in contact with the lamina at \(E, F\), the mid-points of \(DC, CB\) respectively. It may be supposed that the plane of the lamina is kept vertical by smooth constraints. A force \(P\), parallel to \(DB\), is applied at \(A\) and is increased from zero, and equilibrium is ultimately broken. If this occurs by the lamina beginning to rotate about \(F\), show that the coefficient of friction at \(F\) is at least \(\frac{1}{2}\). If, on the other hand, equilibrium is broken by slipping at \(E\) and \(F\), where the coefficients of friction are \(\mu_1, \mu_2\) respectively, show that, when slipping occurs, the ratio of the normal reaction at \(E\) to that at \(F\) is \((1-2\mu_2):(1+2\mu_1)\). Show also that in these circumstances the value of \(P\) is then \[ \frac{W(\mu_1+\mu_2)}{2+\mu_1-\mu_2+4\mu_1\mu_2}. \]
A circular cylinder of radius \(a\) is fixed with its axis horizontal. On the cylinder rests a thick plank with its length horizontal and perpendicular to the generators and its centre of gravity at a height \(b\) vertically above the highest generator. The surfaces are sufficiently rough to prevent slipping. Show that the plank is in stable or unstable equilibrium according as \(b< a\) or \(b \ge a\).
A gun, situated on level ground, is firing at a vehicle which is moving directly away from the gun with velocity \(V\). At the instant of firing the vehicle is distant \(l\) from the gun, and at the same level. Assuming that the vehicle is within range and that \(V\) is small compared with the speed \(U\) of projection, show that on account of the motion of the vehicle the elevation of the gun should be increased from \(\theta_0\) (corresponding to a stationary target) to \(\theta_0+(V/U)\sin\theta_0\sec 2\theta_0\), approximately, provided that \(\theta_0\) is not nearly equal to \(\frac{1}{4}\pi\). Comment on this restriction. Show also that the time of flight of the projectile is greater by \((V/g)\tan 2\theta_0\), approximately, than the value corresponding to a stationary target.
A particle of mass \(m\) is suspended from a fixed support by a light elastic string. When the mass is in equilibrium the extension of the string is \(c\). Show that, when the extension of the string is \(x\), the energy stored in the string is \(\frac{1}{2}m\omega^2 x^2\), where \(\omega^2=g/c\), and prove that, for vertical oscillations (in which the string does not become slack) about the position of equilibrium, the length of the equivalent simple pendulum is \(c\). If the mass is pulled vertically downwards a distance \(kc\) (\(k>1\)) below the position of equilibrium and released from rest, show that the mass first reaches the highest point of its path in a time \[ \{\pi + \text{cosec}^{-1} k + (k^2-1)^{\frac{1}{2}}\}/\omega. \] It may be assumed that \(k\) is such that the mass does not rise as high as the support.
A rocket is travelling vertically upwards. Its initial mass is \(M\), and a mass of gas \(q\) per unit time is discharged at a constant rate vertically downwards with constant velocity \(V\) relative to the rocket. By considering the change of momentum in a small interval of time \(dt\), show that the equation of motion is \[ m\frac{dv}{dt} = qV-mg, \] where \(m\) is the mass of the rocket and \(v\) its velocity at a time \(t\). Deduce that, if the rocket starts from rest at ground level at \(t=0\), \[ v=V\log\{M/(M-qt)\}-gt. \]
A small bead, of mass \(m\), slides on a smooth wire bent in the form of a parabola, of which the plane is vertical and the vertex is at the lowest point. If the bead is released from rest at one end of the latus rectum, find the reaction of the wire on the bead when the bead is at the vertex.
A plane lamina is moving in its own plane. Show that at any instant the lamina is in general rotating about a point (the "instantaneous centre"). One end \(A\) of a straight connecting-rod \(AB\), of length \(l\), is made to describe a circle, of centre \(O\) and radius \(a\) (\(< l\)); the other end \(B\) is constrained to remain on a straight line \(Ox\) in the plane of the circle. Show in a diagram the position of the instantaneous centre \(I\) of \(AB\) for a general position of \(A\). If \((x,y)\) are the coordinates of \(I\) relative to rectangular axes \(Ox, Oy\) in the plane of the circle, show that \[ (x^2+y^2)(x^2-l^2+a^2)^2=4a^2x^4. \]