Two equal uniform circular cylinders each of weight \(w\) rest on a rough horizontal plane with their axes parallel. They support in a symmetrical position a uniform isosceles triangular prism of weight \(W\) with its base uppermost. If \(\lambda\) is the angle of friction between either cylinder and the prism, \(\lambda'\) the angle of friction between either cylinder and the plane, and each base angle of the prism is \(\alpha\), prove that for equilibrium to be possible \(\lambda \ge \alpha\) and \((W+2w)\tan\lambda' \ge W\tan\alpha\). When the system is in equilibrium a gradually increasing force is applied at the centre of the base of the prism and acting vertically downwards. Show that, if \(\lambda' > \frac{\alpha}{2}\) equilibrium will not be broken, but that, if \(\lambda' < \frac{\alpha}{2}\) equilibrium will be broken by the cylinders slipping on the plane.
One end \(A\) of a uniform rod \(AB\) of mass \(m\) and length \(c\) is freely pivoted, and the end \(B\) is connected by a light inextensible string of length \(a\) to a point \(C\) vertically above \(A\). A particle of mass \(M\) is attached to the mid-point of the string. In the equilibrium position \(AB\) and the upper half of the string make acute angles \(\theta, \phi\) respectively with the downward vertical, and the string is entirely above the rod. By use of the potential energy function, or otherwise, prove that \[ (m+2M)b \sin\theta \sin\phi = (Ma\sin\phi + mc\sin\theta)\sin(\theta-\phi), \] where \(b\) is the height of \(C\) above \(A\).
The figure represents a crane supported at two points \(A, B\) in the same horizontal line. In comparison with the load \(w\) and counterpoise \(W\) the weights of the other parts may be neglected. The lengths of \(BC\) and \(DC\) are equal and \(BD\) is equally inclined to \(BA\) and \(BC\). The lengths in feet of other members are indicated. Calculate the ratio \(\frac{W}{w}\) in each of the cases when (i) the reactions at \(A\) and \(B\) are equal, (ii) the stresses in \(AD\) and \(BD\) are equal.
\(A\) and \(C\) are the ends of an unstretched light elastic string of length \(a\) which is lying on a horizontal table. One particle of weight \(w\) is attached at \(C\) and an equal particle at \(B\), where \(B\) divides the string in the ratio \(p:q\). If the system is now freely suspended in equilibrium from the end \(A\), it is observed that \(B\) becomes the mid-point of the string. Find in terms of \(w, p, q\) the modulus of elasticity of the string. If \(A\) remains at the same horizontal level, show that the potential energy in the second position is less than in the first position by an amount \[ \frac{wa(4p-q)}{4p-2q}. \] Discuss the case \(q=2p\).
For a particle moving freely under gravity prove that, if it is possible to project the particle from a given point \(O\) with given speed \(V\) so as to pass through a given point \(P\), then there are in general two possible directions of projection. If these two directions are perpendicular, show that \(P\) must lie on the parabola whose vertex is \(O\) and focus vertically below \(O\) at a depth \(\frac{V^2}{4g}\), and that the difference in the times taken in the two paths cannot be less than \(\frac{V}{g}\sqrt{2}\).
A hollow circular cylinder of internal radius \(a\) is fixed with its axis horizontal. A particle is projected from a point on the lowest generator and moves initially on the smooth inner surface of the cylinder in a plane at right angles to the axis. Find the velocity of projection in order that the particle shall leave the cylinder and pass through the axis.
Obtain expressions for the tangential and normal components of acceleration of a particle moving in a plane. A particle moves on a uniformly rough inclined plane. Initially its speed is \(u\) in a horizontal direction. When the speed is a minimum its direction makes an angle \(\frac{\pi}{3}\) with the downward line of greatest slope. Show that the minimum speed is approximately \(0.88u\).
A smooth sphere impinges obliquely on an equal smooth sphere at rest. Find, in terms of the coefficient of restitution \(e\), the greatest possible deflection undergone by the first sphere, and show that in this case the kinetic energy of the system is reduced by the impact in a ratio which cannot be less than \(4\sqrt{2}-5\) whatever the value of \(e\).
For a lamina in motion in its own plane define the instantaneous centre \(I\), and prove that the motion of the lamina can be reproduced by rolling the locus of \(I\) in the lamina (the body centrode) on the locus of \(I\) in space (the space centrode). Find the instantaneous centre for the motion of the rod \(AB\) in which the end \(A\) is moving along the given line \(CA\) and the end \(B\) at a constant distance from the given point \(C\).
A uniform rod \(AB\) of length \(2a\) can turn without friction about the end \(A\) in a vertical plane. A light elastic string of natural length \(a\) connects the end \(B\) to the point \(C\) vertically above \(A\) such that the length of \(AC\) is \(2a\). When the system is in equilibrium \(ABC\) is an equilateral triangle. Prove that the period of small oscillations about this equilibrium position is \(2\pi\sqrt{\frac{8a}{3g}}\).