A uniform ladder of weight \(w\) and length \(2l\) is placed with one end on the ground and the other end against a vertical wall. The ground slopes upwards towards the wall with which it makes an angle \(\frac{\pi}{2}+\alpha\) and the ground and wall are equally rough, \(\lambda(>\alpha)\) being the angle of friction. The ladder makes angles \(\theta\) and \(\frac{\pi}{2}-\alpha-\theta\) with the wall and ground respectively. \par Shew that the greatest distance up the ladder which a man of weight \(W\) may ascend without causing the ladder to slip is \[ \frac{l[2(w+W)\sin(\lambda-\alpha)\cos(\theta-\lambda)-w\sin\theta\cos\alpha]}{W\sin\theta\cos\alpha}. \]
Explain the use of Bow's notation in graphical statics. \par The diagram represents a pin-jointed framework of light rods held at two points \(A\) and \(O\) in the same vertical line. Determine by graphical methods the stress in each rod and the angle which the reaction at \(O\) makes with \(OA\).
An elastic string \(OA\), of mass \(m\) and coefficient of elasticity \(\lambda\), has when unstretched a length \(l\) and uniform line density. The string hangs in equilibrium from \(O\). Prove that the total extension of the string is \(\frac{mgl}{2\lambda}\). Shew further that the potential energy of the string is less by an amount \(\frac{mgl}{2}+\frac{m^2g^2l}{6\lambda}\) than when it is coiled up at \(O\).
A particle is projected with velocity \(V\) at an angle \(\alpha\) to the horizontal. Prove that its path is a parabola and find the length of its latus rectum. \par Two particles are projected simultaneously with velocities \(V, V'\) at angles \(\alpha, \alpha'\) with the horizontal from the same point \(O\) and in the same vertical plane. \(\alpha>\alpha'\) and \(V'\cos\alpha' > V\cos\alpha\). Prove that the particles arrive at the points of contact of the common tangent to the two trajectories at the same instant \(t\), where \[ t = \frac{VV'\sin(\alpha-\alpha')}{g(V'\cos\alpha'-V\cos\alpha)}. \]
Two particles \(A\) and \(B\) are in motion in a plane. Explain how to find the velocity of \(B\) relative to \(A\) in terms of the velocities of \(A\) and \(B\) referred to fixed axes in the plane. \par An aeroplane is flying at a uniform height with constant velocity \(v\). It is circling about a ship which is moving in a straight line with constant velocity \(ev\) where \(e<1\). Prove that the time taken to describe one such circle is \[ \int_0^{2\pi} \frac{a}{v(1-e^2)}\sqrt{1-e^2\sin^2\theta}\,d\theta, \] where \(a\) is the radius of the circle.
A particle is moving in a straight line so that \[ (2ksv^2+1)^3 = (3ktv^3+1)^2, \] where \(v\) is the velocity and \(s\) the distance described, both measured at the instant \(t\). \(k\) is a constant. Find the acceleration in terms of \(v\). \par If \(v=u\) when \(t=0\), shew that \(v=2u\) when \(s=\frac{3}{8ku^2}\).
When a body is immersed in liquid it is acted upon by an upward vertical force equal to the weight of liquid displaced and passing through the centroid of the volume immersed. Consider the case of a solid circular cylinder of radius \(a\) partly immersed in liquid in a hollow circular cylinder of radius \(b\). Assuming that the solid cylinder does not come in contact with the base of the hollow cylinder and that it is always partly immersed, shew that the motion of the solid cylinder is simple harmonic with period \(2\pi\sqrt{\frac{l(b^2-a^2)}{gb^2}}\), where \(l\) is the length of the solid cylinder immersed when floating in equilibrium. It may be assumed that the axes of both cylinders are vertical.
A particle is suspended from a fixed point by a light inextensible string of length \(l\). If the particle receives a horizontal velocity \(u\), find conditions such that the string shall become slack in the subsequent motion, and prove that in this case the string is slack for a time \(\sqrt{\frac{8l}{g}}\sin\phi \sin 2\phi\) where \(3\cos\phi = \frac{u^2}{gl}-2\).
A uniform straight rod at rest receives simultaneously an impulse \(P\) in the direction of its length and an impulse \(Q\) at one end in a direction perpendicular to the rod. If the initial velocities of the two ends are in perpendicular directions, determine the relation between \(P\) and \(Q\).