Each of six similar particles is of weight \(w\), and is attached to a point \(O\) by a light inextensible string of length \(l\). Any two particles repel each other with a force equal to \(\lambda\) times their distance apart (\(\lambda\) is greater than \(w/6l\)). Prove that when the particles are in equilibrium at the vertices of a regular hexagon, the length of a side of the hexagon is \[ (l^2 - w^2/36\lambda^2)^{\frac{1}{2}}. \]
A uniform rectangular block, whose edges are of length \(2a, 2b, 2c\), and whose weight is \(w\), rests in equilibrium on a rough plane inclined to the horizontal at an angle \(\alpha\). The edges of length \(2c\) are horizontal, and those of length \(2a\) are parallel to a line of greatest slope. If \(\mu\) is the coefficient of friction, shew that \(\mu > \tan\alpha\) and \(b\tan\alpha < a\). A string is attached to the middle of the lower horizontal edge of the face parallel to, but not in contact with, the plane. It is parallel to a line of greatest slope, and passes over the middle of the upper horizontal edge. The tension in the string is gradually increased from zero. Shew that the block slips or tilts when equilibrium is first broken according as \[ a \lesseqgtr b(2\mu + \tan\alpha). \]
State necessary and sufficient conditions for a system of forces in one plane to be in equilibrium. Forces whose magnitudes are proportional to the sides of a triangle act inwards perpendicular to the sides at their middle points. Prove that the system is in equilibrium. If the lines of action of the forces be all rotated in the same sense through the same angle, each about the point where it meets its side of the triangle, shew that the system is equivalent to a couple, and find its magnitude.
Four uniform rods, each of length \(a\) and weight \(w\), are smoothly jointed together to form a rhombus \(ABCD\). The rhombus is kept in shape by a light strut \(BD\), and hung from \(A\). A uniform circular disc of radius \(r\) and weight \(w'\) is placed inside the triangle \(BCD\) with its plane vertical, and its smooth rim in contact with \(BC\) and \(DC\); its highest point is below \(BD\). If the angle \(BAC\) is \(\theta\), shew that the thrust in \(BD\) is \[ (2w+w')\tan\theta - w'\frac{r}{\sin^2\theta\cos\theta}. \]
When the velocity of a train of mass \(M\) lb. is \(v_0\) feet per second, it starts picking up water at a constant rate \(M/a\) lb. per second. If the engine works at constant power, shew that if all frictional resistances were neglected the velocity after \(t\) seconds would be \[ \{v_1^2 - a^2V^2/(t+a)^2\}^{\frac{1}{2}} \text{ feet per second}, \] where \(v_1\) feet per second is the limiting velocity, and \(V^2 = v_1^2 - v_0^2\).
A heavy particle is attached to a fixed point \(O\) by a light elastic string of natural length \(l\). When the string is vertical and the system at rest, the length of the string is \(l/\nu\). If the particle describes a horizontal circle about the vertical through \(O\) with uniform angular velocity \((g\kappa/l)^{\frac{1}{2}}\), prove that the radius of the circle is \[ l\{v^2\kappa^2 - (\nu + \nu\kappa - \kappa)^2\}^{\frac{1}{2}}/\kappa(\nu + \nu\kappa - \kappa), \] and that \(\kappa\) must lie between \(\nu\) and \(\nu/(1-\nu)\).
A particle is projected with a given velocity from a given point in a horizontal plane, so that, at a horizontal distance \(d\) from the point of projection, it just passes over the highest post that any such projectile could pass over. Shew that the height of the projectile at a horizontal distance \(\frac{1}{2}d\) from the point of projection is greater by \(\frac{1}{8}R\) than a quarter of the height of the post, where \(R\) is the maximum range on the horizontal plane.
A sphere collides obliquely with another sphere of equal mass, which is at rest, both spheres being smooth and perfectly resilient. Shew that after the collision their paths are at right angles. \(\alpha, \beta, \gamma\) are three smooth, perfectly resilient spheres with the same radius, 3 cm., and the same mass, and are on a smooth horizontal table. The centres of \(\beta\) and \(\gamma\) are at \(B\) and \(C\), where \(BC = 16\) cm. \(\alpha\) is projected with velocity \(V\) at right angles to \(BC\), and strikes first \(\beta\) and then \(\gamma\). Its final path is at right angles to \(BC\). If it strikes \(\beta\) at \(P\), find the cosine of the angle \(PBC\), and shew that the final velocities of \(\alpha, \beta, \gamma\) are \(\frac{9V}{25}, \frac{4V}{5}\) and \(\frac{12V}{25}\) respectively.
A particle of mass \(M\) is hung from two strings, each of length 12 feet, whose other ends are attached to two rings, each of mass \(\frac{1}{2}M\), which slide on a smooth horizontal rod. The system is released from rest with the strings taut, and the particle level with the rod and between the rings. Shew that when the particle has descended a distance 8 feet, its velocity is 16 feet per second. [Take \(g=32\) feet per second per second.]
A particle of mass \(m\) is attached to the four corners of a square, whose diagonal is of length \(2a\), by four elastic strings each of length \(a\), stretched under a tension \(P\). Determine the period of a small oscillation of the particle for displacements normal to the plane of the square, and shew that for displacements along a diagonal of the square the period is \[ 2\pi\left(\frac{ma}{4P+2\lambda}\right)^{\frac{1}{2}}, \] where \(\lambda\) is the tension in a string when its length is twice the unstretched length.