Four bars are freely jointed at their ends so as to form a plane quadrilateral \(ABCD\), and the opposite corners are joined by tight elastic strings \(AC, BD\). The whole lies on a smooth horizontal plane. If \(T, T'\) are the tensions of the strings and \(O\) is the point where they cross, prove that \[ \frac{T \cdot AC}{OA \cdot OC} = \frac{T' \cdot BD}{OB \cdot OD}. \]
A uniform square lamina has a fine inextensible string of length equal to that of one side attached to one corner. The other end of the string is attached to a fixed point on a rough vertical wall, and the lamina rests with its plane vertical and perpendicular to the wall. Prove that if the coefficient of friction is unity, the angle which the string makes with the wall lies between \(\dfrac{\pi}{4}\) and \(\tan^{-1}\dfrac{1}{2}\).
A particle \(P\) of mass \(m\) rests on a rough horizontal table whose coefficient of friction is \(\mu\), and is attached to one end of a fine inextensible string which passes over a smooth fixed pulley \(A\) at the edge of the table. The string then passes under a smooth movable pulley \(B\) of mass \(m\) and over a smooth fixed pulley \(C\), the other end of the string being attached to a particle \(D\) of mass \(m\) which hangs vertically. All the portions of the string not in contact with a pulley are horizontal or vertical. Prove that, if \(\mu > \frac{3}{2}\), \(P\) will not move, and that, if \(\mu < \frac{3}{2}\), \(D\) will move with acceleration \((3-2\mu)g/6\).
The relative velocity of the ends \(H\) and \(M\) of the hour and minute hands of a watch is calculated (i) relatively to the face, and (ii) relatively to the seconds hand. Prove that the values obtained are different, their vector difference being \(\dfrac{\pi a}{30}\) feet per second perpendicular to \(HM\), if \(a\) feet is the length of \(HM\). Give a definition of relative velocity consistent with this result.
A particle moves straight along the smooth interior of a straight tube which itself is moving in the direction of its own length on a smooth horizontal table, both ends of the tube being closed. If \(e\) is the coefficient of restitution, prove that the kinetic energy lost in consecutive impacts diminishes in the ratio \(e^2\), and that the time between consecutive impacts increases in the ratio \(\dfrac{1}{e}\).
Two masses, \(m_1\) and \(m_2\) lb., are connected by a light elastic string passing over a smooth pulley. The string stretches one foot under a tension of \(P\) poundals. The masses are supported so that the two sides of the string are vertical and just slack, and the mass \(m_1\) is then released. Prove that the mass \(m_2\) will begin to rise after a time \[ \sqrt{\frac{m_1}{P}}\cos^{-1}\left(1-\frac{m_2}{m_1}\right). \]
The plane of a parabola is vertical and its axis is inclined at an angle \(3\alpha\) (\(a < \frac{\pi}{6}\)) to the vertical and its vertex is upwards. A particle is to slide down a smooth straight line under gravity from the curve to the focus. Prove that the time of descent is a minimum when the line is inclined at an angle \(\alpha\) to the vertical. Prove also that the chord of quickest descent from the focus to the curve makes an angle \(\dfrac{\pi}{3}-\alpha\) with the vertical.
Prove that the length and area of the loop of the curve \(3ay^2=x(x-a)^2\) are \(\dfrac{4a}{\sqrt{3}}\) and \(\dfrac{8a^2}{15\sqrt{3}}\) respectively.
A rigid roof-frame \(ABC\) is in the form of an isosceles triangle with a right angle at \(B\), and rests upon two walls at \(A, C\). It carries a weight \(W\) symmetrically distributed on the two sides, and due to wind pressure there is a force \(w\) uniformly spread over \(BC\) and perpendicular to it. If the reaction at \(C\) is vertical, find the horizontal and vertical components of the reaction at \(A\).