A particle can move on a smooth plane inclined at an angle \(\alpha\) to the horizontal and is attached to a point of the plane by a light inextensible string of length \(l\). The particle is at rest in equilibrium when it is given velocity \(V\) in the horizontal direction in the plane. Find the limits within which \(V\) must lie if the string is to become slack during the subsequent motion.
Three light rods \(BC, CA, AB\) each of length \(a\) are jointed together to form an equilateral triangle, which is suspended from a point \(O\) by three equal strings \(OA, OB, OC\) each of length \(b\). A weight \(W\) is suspended below the framework by three strings attached to \(A, B\) and \(C\) each of length \(c\). Find the resulting thrust created in each rod.
Explain what is meant by the angle of friction between two bodies in contact. The faces of a double inclined plane both make the same angle with the horizontal, and a rough heavy uniform chain lies across the ridge and stretches down lines of greatest slope on the two sides of the ridge. Find, in terms of the angle of friction, the inclination to the horizontal of the line joining the two ends of the chain if the system is in limiting equilibrium.
Show that referred to suitable axes the equation of the form in which a uniform heavy chain hangs under gravity is \(y=c\cosh\dfrac{x}{c}\). Obtain also the relation between \(x\) and the arc length \(s\). One end \(A\) of a uniform heavy chain \(AB\) of length \(2l\) is fixed while the other end \(B\) is moved very slowly along a horizontal rail through \(A\). Show that the locus described by the lowest point of the catenary in which the chain hangs, referred to horizontal and vertical (downwards) axes at \(A\), has equation \[ 2xy = (l^2-y^2)\{\log(l+y)-\log(l-y)\}. \]
A uniform perfectly rough plank of thickness \(2b\) rests across a fixed cylinder of radius \(a\) whose axis is horizontal. The plank is in equilibrium with its length inclined at an angle \(\alpha\) to the horizontal, and is rolled on the cylinder so that this angle increases to \(\theta\). Calculate the increase in potential energy of the system and hence show that the original position is one of stable equilibrium if \(a\cos^2\alpha>b\).
A particle of unit mass is allowed to fall from rest under gravity in a medium that produces on it a retardation equal to \(k\) times its velocity, and at the instant of release an equal particle is projected vertically downwards from the same place with initial speed \(v\). Show that their vertical distance apart tends ultimately to the value \(v/k\).
A light string passes over a small smooth fixed pulley and to one end is attached a mass \(M\) and to the other a second small light pulley over which passes a second string carrying masses \(m_1\) and \(m_2\) at its ends. Find the condition that if the system is released from rest the mass \(M\) will not move, and determine the total downward force on the fixed pulley.
Explain what is meant by simple harmonic motion. Derive and solve the differential equation of such motion. A particle is describing simple harmonic motion with period \(2\pi/n\) and its velocities at two points distance \(h\) apart are \(u\) and \(v\). Show that the square of the amplitude of the motion is \[ \tfrac{1}{4}\{h^2 + 2(u^2+v^2)/n^2 + (u^2-v^2)^2/n^4h^2\}. \]
A fire-engine working at a rate of \(E\) horse-power pumps \(w\) cubic feet of water per second from a part of a reservoir at depth \(d\) feet below the open end of the hose. If the hose is held at an angle \(\alpha\) to the horizon, find the maximum height that the resulting jet of water can reach. 1 H.P. = 550 ft.-lb. per sec., 1 cu. ft. of water weighs 62.5 lb.
Three particles each of mass \(m\) are situated instantaneously at the vertices \(A,B,C\) of a triangle and are moving in such a way that after time \(t\) they all collide simultaneously at a point \(P\) in the plane \(ABC\) and coalesce to form a single particle of mass \(3m\). Show that the kinetic energy lost is \(m(BC^2+CA^2+AB^2)/6t^2\).