Problems

A thin wire has the form of a circle in a vertical plane with centre \(C\). \(A, B\) are pegs attached to the wire so that \(CA, CB\) make angles \(\alpha\) on opposite sides of the downward vertical through \(C\). A small ring of mass \(M\) can slide on the wire, and is attached to two strings passed over the pegs with masses \(m\) hanging from their ends. Write down the potential energy of the system when the radius to \(M\) makes an angle \(\theta\) with the vertical. Hence discuss the stability of equilibrium positions in the cases \[ M \gtreqqless m\sin\alpha. \]

A particle of mass \(m\) is attached by a string to a point on the circumference of a fixed circular cylinder of radius \(a\) whose axis is vertical, the string being initially horizontal and tangential to the cylinder. 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 in the string is inversely proportional to the length which remains straight at any moment, (iii) that if the initial length of the string is \(l\) and the greatest tension the string can bear is \(T\), the string will break when it has turned through an angle \[ l/a-mv^2/aT. \]

Shew that if a number of particles connected by inelastic strings move under no forces, their linear momentum and energy are constant. Three equal particles \(A, B, C\) connected by inelastic strings \(AB, BC\) of length \(a\) lie at rest with the strings in a straight line on a smooth horizontal table. \(B\) is projected with velocity \(V\) at right angles to \(AB\). Shew that the particles \(A\) and \(C\) afterwards collide with relative velocity \(\frac{2V}{\sqrt{3}}\). If the coefficient of restitution is \(e\), find the velocities of the three particles when the string is again straight.

Explain and illustrate the principle of duality in projective geometry, and discuss the bearing on this principle of the theory of reciprocation with respect to a conic.

\(ABC\) is a triangle, \(O\) the centre of its circumcircle. Forces \(P,Q,R\) act along \(BC, CA, AB\), and forces \(P',Q',R'\) along \(OA, OB, OC\). Shew that if the forces are in equilibrium \[ P\cos A + Q\cos B + R\cos C = 0, \] and \[ \frac{PP'}{\sin A} + \frac{QQ'}{\sin B} + \frac{RR'}{\sin C} = 0. \]

\(A_1, A_2, A_3, A_4\) are four coplanar points such that the line joining any two is perpendicular to the line joining the other two; \(S_1, S_2, S_3, S_4\) are the circumcentres of the triangles \(A_2A_3A_4, A_3A_4A_1, A_4A_1A_2, A_1A_2A_3\). Prove that \(A_1\) is the circumcentre of \(S_2S_3S_4\), and that all the eight triangles whose vertices are either three of the \(A\)'s or three of the \(S\)'s have the same nine point circle. Shew also that the quadrangles \(A_1A_2A_3A_4\) and \(S_1S_2S_3S_4\) are congruent.

\(ABC\) is a triangle obtuse-angled at \(A\); \(D\) is the foot of the perpendicular from \(A\) on the side \(BC\). If \(BD\) equals \(3AD\), and \(CD\) equals \(2AD\), prove by calculations based on geometrical theorems that the angle \(BAC\) is \(135^\circ\).

Investigate the correspondence between points in a plane defined by \[ x' : y' : z' :: a_1x+b_1y+c_1z : a_2x+b_2y+c_2z : a_3x+b_3y+c_3z, \] where the determinant of the coefficients does not vanish. Prove in particular that straight lines correspond to straight lines; that the cross ratio of any four points is equal to that of the four corresponding points; that there are in general only three but there may be a whole line of double points; and that it is possible to assign arbitrarily four pairs of corresponding points, subject to a restriction about collinearity. Prove also that it is possible to choose the coefficients so that any two arbitrarily assigned non-degenerate conics correspond, and reconcile this with the impossibility of choosing arbitrarily five pairs of corresponding points.

\(ABCDEFGH\) is an octagon composed of eight similar uniform rods, each of weight \(w\), freely hinged together. Shew that it can be supported as a regular octagon in a vertical plane with \(AE\) vertical if certain vertical forces are applied at the angular points other than \(A\) and \(E\); sketch the force diagram and determine these forces.

\(p,q,r,s\) are the four common tangents to two conics \(S\) and \(S'\). The points of contact of \(p\) are \(P, P'\); those of \(q, Q, Q'\); \(r\) meets \(p\) and \(q\) in \(A\) and \(B\); \(s\) meets \(p\) and \(q\) in \(C\) and \(D\). Prove that the cross ratios \((PAP'C)\) and \((QDQ'B)\) are equal; and that if \(PP'\) divides \(AC\) harmonically and \(r,s\) meet in \(E\), \(E\) is conjugate to \(P'\) with respect to \(S\) and to \(P\) with respect to \(S'\).