A lamina is in equilibrium under the joint action of two systems of forces in its plane, all of given magnitudes and applied at given points. All the forces of the first system are then turned anticlockwise through an angle \(\theta\) about their respective points of application, and all those of the second system are turned clockwise through the same angle \(\theta\). Show that in general the resultant of the new set of forces is a single force whose line of action is independent of \(\theta\); but that if, and only if, each of the two original systems, taken separately, reduces to a couple or is in equilibrium, then the resultant is a couple (or exceptionally the new set of forces may be in equilibrium).
The ends of a uniform rod of length \(8a\) are free to move on a fixed smooth wire bent in the form of a parabola of latus rectum \(4a\) with axis vertical and vertex downwards. Express the potential energy of the rod in terms of its inclination to the horizontal; find the inclinations of the rod to the horizontal in all the possible positions of equilibrium and determine whether each position is stable or unstable. Show that the period of small oscillations about a stable position of equilibrium is \[ 2\pi\sqrt{\left(\frac{1a}{3g}\right)}. \]
The centre of mass of a car, moving in a straight line on level ground, is at height \(h\) above ground level and at a distance \(a\) from the vertical plane through the rear axle and \(b\) from the vertical plane through the front axles. Show that if braking is applied equally to the two rear wheels only, excessive braking may cause a skid but cannot cause the wheels to leave the ground; but that if braking is applied to the front wheels the rear wheels may leave the ground if the coefficient of friction \(\mu\) is great enough; and find the condition for this to happen, neglecting the rotatory inertia of the revolving parts. If the braking force is divided between the front and back wheels, determine whether it is possible to get more effective braking than with front-wheel brakes only, and whether the result is any different if the car is running downhill. If the angular momentum of all rotating parts may be assumed to be in the same direction as that of the wheels, and proportional to the car's speed, determine whether the maximum attainable retardation is greater or less than if this angular momentum were negligible.
A lamina is moving in any manner in a plane. The coordinates of a point \(P\) fixed in the lamina are \((X,Y)\) with respect to axes with origin \(O\) fixed in space, and \((x,y)\) with respect to axes, with origin \(O'\), fixed in the lamina. The velocity of \(O'\) has components \((u,v)\) parallel to the \((X,Y)\) axes and the \(x\)-axis makes an angle \(\theta\) with the \(X\)-axis, \(u,v\) and \(\theta\) being given functions of time. Determine the components parallel to the \((X,Y)\) axes of the velocity and acceleration of \(P\) in terms of \(x,y\) and \(u,v\) and \(\theta\) and their time derivatives. Show that in general the points where the acceleration is perpendicular to the velocity at any given instant lie on a circle. \(A\) and \(B\) are two points fixed in the lamina distant \(l\) apart, and they are constrained to move along \(OX\) and \(OY\) respectively, their displacements from \(O\) at time \(t\) being \(l\sin nt\) and \(l\cos nt\). Show that at any instant the points where the velocity is perpendicular to the acceleration lie on the straight line joining \(O\) to the instantaneous centre of rotation.
A wedge of mass \(m\) with its two faces inclined at an angle \(\pi/3\) is at rest on a horizontal plane, the face in contact with the plane being perfectly smooth. It is hit by a uniform right circular cylinder of mass \(m\) and radius \(a\) which is initially rolling along the horizontal plane with the velocity \(a\Omega\), its axis being parallel to the line of intersection of the two faces of the wedge. Assuming that there is no rebound or slipping on impact, show that immediately after the impact, the velocity of the wedge is \(4a\Omega/11\). The cylinder then rolls up the inclined face and back to the horizontal plane; show that the velocity of the wedge when the cylinder reaches the horizontal plane again is \(7a\Omega/11\), and find the final velocity of the cylinder on the horizontal plane, assuming that there is no rebound or slipping (at the point of contact with the plane) on the final impact.
The mid-points of the sides \(AB, CD\) of a parallelogram \(ABCD\) are \(X, Y\). \(P\) is a point on the diagonal \(AC\) and \(PX, PY\) meet \(BC, AD\) respectively in \(U, V\). Prove that \(UV\) is parallel to \(AB\). If \(UV\) cuts \(AC\) in \(W\), prove that \(PW\) is the harmonic mean of \(PA\) and \(PC\), due regard being paid to sign.
Show that by suitable choice of homogeneous coordinates a non-singular conic \(S\) can be expressed in the parametric form \[ x:y:z=t^2:t:1. \] The lines joining the point \(Y(0,1,0)\) to four points \(P_1, P_2, P_3, P_4\) of \(S\) meet \(S\) again in \(Q_1, Q_2, Q_3, Q_4\). Prove that the conics \(YP_1P_2P_3P_4\) and \(YQ_1Q_2Q_3Q_4\) touch at \(Y\), and find, in terms of the parameters of \(P_1, P_2, P_3, P_4\) on \(S\), a necessary and sufficient condition that these conics should also touch at a point other than \(Y\).
Define the greatest common factor of two integers \(m, n\) and describe a method of determining it. The integers \(u_0, u_1, \dots, u_n, \dots\) are defined by \[ u_0=0, \quad u_1=1, \quad u_{r+1}=u_r+u_{r-1} \quad (r \ge 1). \] Prove that, if \(r, s\) are positive integers \[ u_{r+s} = u_r u_{s+1} + u_s u_{r-1} \] and that, if \(s < r\), \[ u_{r-s} = (-1)^s (u_r u_{s+1} - u_s u_{r+1}). \] Deduce that the greatest common factor of \(u_{207}\) and \(u_{345}\) is \(u_{69}\).
Prove that, if \(i^2=-1\) and \(n\) is a positive integer, \[ \left(\frac{1+i\tan\theta}{1-i\tan\theta}\right)^n = \frac{1+i\tan n\theta}{1-i\tan n\theta}. \] Hence, or otherwise, express \(\tan n\theta\) in the form \(\dfrac{f(\tan\theta)}{g(\tan\theta)}\), where \(f(x)\) and \(g(x)\) are polynomials in \(x\). Evaluate \[ \text{(i) } \sum_{k=0}^{k=n-1} \cot \frac{(4k+1)\pi}{4n}; \quad \text{(ii) } \sum_{k=0}^{k=n-1} \cot^2 \frac{(4k+1)\pi}{4n}. \]
Defining \(\log_e t = \int_1^t \frac{du}{u}\), prove that, for \(t>0\), \(\log_e t < t-1\). Let \(f(x ) > 0\) for \(0 < x < 1\). By considering first the special case when \[ \int_0^1 f(t)dt = 1, \] prove the inequality \[ \int_0^1 \log_e f(t)dt \le \log_e \left\{\int_0^1 f(t)dt\right\}. \]