Two lines \(ABC\dots\), \(A'B'C'\dots\) meet in a point \(O\). Shew that forces acting along \(AA'\), \(BB'\), \(CC'\), \dots, of magnitudes \(\lambda OA \cdot OA'\), \(\mu OB \cdot OB'\), \(\nu OC \cdot OC'\), \dots respectively, are in equilibrium, if \(\sum \lambda OA = 0\), \(\sum \lambda OA' = 0\), \(\sum \lambda OA \cdot OA' = 0\).
A truck has four wheels and the distance between the two axles is \(2a\); the centre of gravity is midway between the axles and is at a perpendicular distance \(h\) from the ground. The truck just slips down a slope of angle \(\theta\) when the lower wheels alone are locked; prove that \(2a > h\mu\) and \[ \tan\theta = \frac{\mu a}{2a-h\mu}, \] where \(\mu\) is the coefficient of friction. Find the angle of slope, if the truck just slips when the upper wheels alone are locked.
A circular cylinder of weight \(W\) rests between two equally rough planes, each inclined at an angle \(\alpha\) to the horizontal and intersecting in a horizontal line which is vertically below the axis of the cylinder. A couple \(G\) is applied about the axis of the cylinder and is just sufficient to cause the cylinder to rotate without rolling up either plane. Shew that the coefficient of friction cannot exceed \(\tan\alpha\) and find the value of \(G\).
A weight is suspended by two strings, each of natural length 24 in., from two points 24 in. apart on the same level. The strings have different coefficients of elasticity and are stretched by the weight to lengths 25 and 27 in. Find (with the aid of tables) the ratio of the coefficients of elasticity of the two strings.
A smooth horizontal bar is parallel to a smooth vertical wall and at a distance \(a\) from it. A uniform rod \(AB\), of length \(2l\) (\(l > a\)), is placed over the bar in a plane perpendicular to the wall with \(B\) against the wall. If \(\theta\) is the angle between the rod and the wall, prove that \(\sin^3\theta = a/l\). Shew that, whatever weight is hung from the end \(A\), the angle between the rod and the wall in equilibrium cannot be less than \(\phi\), where \(\sin^3\phi = a/2l\).
The position of a point moving in two dimensions is given by polar coordinates \(r, \theta\); find the component velocities and accelerations along and perpendicular to the radius vector. The velocities of a particle along and perpendicular to a radius vector from a fixed origin are \(\lambda r^2\) and \(\mu \theta^2\); find the polar equation of the path and the component accelerations in terms of \(r\) and \(\theta\).
Find the least velocity \(u\) with which a particle must be projected from a point on the ground so that it may pass over a wall of height \(h\) at a distance \(a\), measured horizontally and perpendicularly to the wall. Find also the velocity \(u'\) with which it then reaches the wall. If the particle is projected with velocity \(v>u\), prove that the length of the top of the wall within range is \[ 2\{(v^2-u^2)(v^2+u'^2)\}^{1/2}/g. \]
Find an expression for the kinetic energy of \(n\) particles of masses \(m_i\) (\(i=1,2,\dots,n\)) moving in a plane in terms of the velocity \(V\) of their centre of mass \(G\) and of the velocities of the particles relative to \(G\). Prove also that the kinetic energy is equal to \[ \tfrac{1}{2}\{\Sigma m_i V^2 + \Sigma m_i m_j v_{ij}^2 / \Sigma m_i \}, \] where \(v_{ij}\) is the velocity of \(m_i\) relative to \(m_j\).
Two particles of masses \(m\) and \(m'\) are connected by a light string passing over a small smooth peg, and are held at rest with the string tight, the vertical lengths of string being \(a\) and \(b\). They are then projected horizontally with velocities \(u\) and \(v\) respectively. Prove that the initial tension in the string is \[ \frac{mm'}{m+m'} \left( \frac{u^2}{a} + \frac{v^2}{b} + 2g \right). \]
Two particles of masses \(4m, 3m\) connected by a taut light string of length \(\frac{1}{2}\pi a\) rest in equilibrium on a smooth horizontal cylinder of radius \(a\). If equilibrium is slightly disturbed so that the heavier particle begins to descend, find at what point it will leave the surface, and shew that at that moment the pressure on the other particle is slightly greater than two-thirds of its weight. [Trigonometrical tables should be used.]