Problems

Shew that a system of coplanar forces is equivalent to a couple if the geometric sum of the forces is zero, i.e. if the same forces acting on a particle would be in equilibrium. The coplanar forces \(P_1, P_2, \dots P_n\), whose geometric sum is not zero, act at points \(A_1, A_2, \dots A_n\) respectively. Shew that if the direction of each force is turned through an angle \(\theta\) the resultant force passes through a point \(C\) for all values of \(\theta\). Shew further that if \(A_2, A_3, \dots A_n\) are fixed, and \(A_1\) moves on a given curve, then \(C\) traces out a similar (but not in general similarly situated) curve. Explain how the exceptional cases (in which the curves are similarly situated) arise.

Two circular cylinders, \(A\) and \(B\), have their axes parallel in the same horizontal plane, \(A\) being fixed and \(B\) free to turn about its axis. A uniform heavy circular cylinder \(C\), having its axis parallel to those of \(A\) and \(B\), is placed in contact with them. The angle that the plane containing the axes of \(A\) and \(C\) makes with the horizontal is denoted by \(\alpha\), and the corresponding angle for \(B\) and \(C\) by \(\beta\). The angle of friction between \(A\) and \(C\) is denoted by \(\theta\), and that between \(B\) and \(C\) by \(\phi\). A couple is applied to \(B\) (in the sense which tends to move the highest point of \(B\) away from \(A\)) and gradually increased. Shew that equilibrium is broken by slipping between \(B\) and \(C\) if \[ \sin\alpha - \sin\beta > \cos\alpha\cot\theta - \cos\beta\cot\phi, \] and that this occurs when the couple attains the value \[ \frac{bW\sin\phi\cos\alpha}{\sin(\alpha+\beta+\phi)+\sin\phi}, \] where \(b\) denotes the radius of \(B\), and \(W\) the weight of \(C\).

Three equal particles \(A, B, C\) rest on a smooth table, \(A\) being joined to \(B\), and \(B\) to \(C\), by tight inelastic strings. The angle between the strings is \(\beta(<\pi/2)\). \(A\) is given a velocity \(u\) in the direction parallel to \(CB\). Shew that when the string \(AB\) again tightens, \(C\) starts off with velocity \(u/(3+4\tan^2\beta)\).

A man of mass \(m\) stands on an escalator of inclination \(\alpha\) which ascends with uniform velocity. He walks up the escalator, and finally comes to rest again relative to the escalator. Shew that if \(a\) is the distance travelled by the escalator, and \(b\) the distance travelled by the man relative to the escalator, then of the total work done \(mga\sin\alpha\) is supplied by the engine and \(mgb\sin\alpha\) by the man. (The man is to be considered as a particle, and the escalator as a continuous gradient without steps. The acceleration of the man relative to the escalator is assumed to be a continuous function of the time.) Consider next the more general problem in which the velocity of the escalator at any instant is \(u\), and of the man relative to the escalator \(v\), \(u\) and \(v\) being functions of the time having continuous derivatives. Shew that of the total work done on the man in any interval of time \(t_1\) to \(t_2\) the amount supplied by the engine is \[ mga\sin\alpha + \frac{1}{2}m(u_2^2-u_1^2) + m\int_{t_1}^{t_2} u\frac{dv}{dt}dt, \] and by the man \[ mgb\sin\alpha + \frac{1}{2}m(v_2^2-v_1^2) + m\int_{t_1}^{t_2} v\frac{du}{dt}dt, \] where \(a\) is the distance travelled by the escalator, and \(b\) by the man relative to the escalator, in the given interval.

A bead of mass \(m\) is free to slide on a smooth horizontal wire. A light rod of length \(a\) is freely attached to the bead, and carries a particle of mass \(m\) at the other end. The system rests in equilibrium, and the particle is struck a blow \(mu\) parallel to the wire. Shew that in the subsequent motion the rod will just become horizontal if \(u=2\sqrt{ga}\). Shew further that if \(u\) has this value the angular velocity of the rod when it makes an angle \(\theta\) with the horizontal is \[ \sqrt{\frac{4g\sin\theta}{a(2-\sin^2\theta)}}. \]

Prove that, if \(ax^2+2bx+c\) is to be positive for all real values of \(x\), it is both necessary and sufficient that

Forces \(P_1, P_2, P_3, P_4, P_5, P_6\) act along the sides of a regular hexagon taken in order. Shew that they will be in equilibrium if \(\Sigma P = 0\) and \(P_1-P_4 = P_3-P_6=P_5-P_2\).

Shew that the four circles which circumscribe the triangles formed by three out of four given lines meet in a point. Hence shew that the five spheres which circumscribe the tetrahedra formed by four out of five given planes meet by fours in five points, one in each plane.

If two circles cut at right angles shew that the intercept made by either circle on any line drawn through the centre of the other circle is divided internally and externally in the same ratio by the points in which it meets the other circle. If the two circles intersect in \(A\) and \(B\) and the diameters of the circles through \(A\) meet them again in \(X\) and \(Y\) shew that the circle through the mid-points of the sides of the triangle \(AXY\) passes through \(A\) and \(B\).

Give an account of the method of Inversion, as applied to plane geometry, shewing its effect upon straight lines, circles and systems of coaxal circles, and proving that magnitudes of angles are unchanged, and that pairs of points inverse with respect to any circle retain the property after inversion. The centre of inversion being the origin, find the equation of the curve inverse to the ellipse \[ 2x = ax^2+by^2+2hxy \] and shew that the asymptote is the inverse of the circle of curvature at the origin. State the property of circles inverse to (i) the circles of curvature of the ellipse, (ii) the circles which touch the ellipse at two points, considering in particular those circles of the families (i) and (ii) which pass through the origin.