When a ship is steaming due N. with a speed \(U\) the wind appears to come from a direction \(\alpha\) E. of N. When the speed is \(2U\) due N. the corresponding direction of the apparent wind is \(\beta\). Show that the wind is blowing from a direction \(\cot^{-1} (2 \cot \alpha - \cot \beta)\) E. of N. Find also the speed of the wind.
Two equally rough fixed planes, each inclined at an angle \(\beta\) to the vertical, have their line of intersection (\(l\)) horizontal. A uniform cube of weight \(W\) rests symmetrically in the trough thus formed, and \(E\) is the centre of the upper horizontal face. A horizontal force \(P\) is applied at \(E\) in a vertical plane perpendicular to \(l\). If, as \(P\) is slowly increased, equilibrium is broken by the cube tilting about one of the lower edges, show that the coefficient of friction \(\mu\) between the cube and the planes must be greater than \[ (\tan \beta - 2)/(1 + 2 \tan \beta), \] provided that \(\tan \beta\) is greater than 2. Give the corresponding result for \(\tan \beta < 2\). If \(\tan \beta > 2\), and \(\mu < (\tan \beta - 2)/(1 + 2 \tan \beta)\), show that the value of \(P\) for which slipping will occur is \[ \mu W/\{1 - (1 + \mu^2) \cos \beta (2 \sin \beta + \cos \beta)\}. \]
A light smoothly-jointed framework in the form of a regular hexagon \(ABCDEF\) is kept rigid by struts \(AC\), \(AD\) and \(AE\). The framework is suspended from \(B\) and a weight 10 lb. is suspended from \(F\). Find the thrust in \(AD\).
A light rod \(AB\) of length \(2a\) is freely pivoted at \(A\) to a point of a vertical wall and carries a particle of mass \(M\) at \(B\). A light spring of modulus \(\lambda\) and unstretched length \(l\) joins the mid-point of \(AB\) to a point of the wall a distance \(h\) vertically above \(A\). If the system is in equilibrium with \(AB\) horizontal, show that \(l = \lambda hs/(2Mgs + \lambda h)\), where \(s^2 = a^2 + h^2\). Show that this position of equilibrium is stable, and find the period of small oscillations about this position.
\(ABC\) and \(ADC\) are two equal uniform thin bars, each weighing \(w\) per unit length and bent at right angles at their mid-points \(B, D\). They are freely jointed at \(A\) and \(C\) to form a square of side \(a\) which hangs at rest from a cord attached at \(A\). Find the bending moment, the shearing force and the tension (i) at a point of \(CB\) distant \(x\) from \(C\), (ii) at a point of \(BA\) distant \(x\) from \(B\).
A particle rests on top of a smooth fixed sphere. If the particle is slightly displaced, find where it leaves the surface. Find also where it crosses the horizontal plane through the centre of the sphere.
A long chain \(AB\) of mass \(\lambda\) lb. per ft. is laid upon the ground in a straight line. The end \(A\) is attached to a motor car which is then driven towards \(B\) with acceleration \(f\) ft. per sec.\(^2\), so that the chain is doubled back on itself. Neglecting friction between the chain and the ground, calculate the tension at \(A\) after \(t\) sec. Show that the kinetic energy of the chain is two-thirds of the work done by the car; explain why these quantities are not equal.
A solid circular drum of radius \(r\) is made from uniform material. Calculate the radius of gyration about its axis. A chain of mass \(m\) is wrapped round a solid circular drum of mass \(M\) and radius \(r\), making \(n\) complete turns, and one end of the chain is fixed to the drum. The drum is mounted on a smooth horizontal axle with a short length of chain left hanging. If the chain is allowed to unwrap itself, and the freely hanging portion descends vertically, find the angular velocity of the drum when half the chain has run out (i) when \(n\) is even, (ii) when \(n\) is odd.
A particle moves in a plane under a force directed towards a fixed point \(O\) and of magnitude \(n^2r\) per unit mass, where \(n\) is a constant and \(r\) is the distance of the particle from \(O\). Initially the particle is at a point \(A\) at a distance \(a\) from \(O\) and has speed \(an\) in a direction making an angle \(\pi/4\) with \(AO\). Prove that the particle describes an ellipse, and find the lengths of the semi-axes.
A stream of particles moving at speed \(v\) falls upon a perfectly elastic plane reflecting surface at an angle of incidence \(\alpha\). If there are \(n\) particles, each of mass \(m\), per unit volume in the incident stream, calculate (i) the number of particles falling on unit area of the surface per unit time, (ii) the force per unit area needed to hold the surface in position. A gas is enclosed in a rectangular box. If there are \(N\) molecules per unit volume, each of mass \(m\), all moving with the same speed \(v\) in directions distributed uniformly in space, show that there are \(\frac{1}{2} N \sin \alpha d\alpha\) molecules per unit volume with velocities making angles lying between \(\alpha\) and \(\alpha + d\alpha\) with any given direction. Calculate the pressure exerted on the sides of the box, assuming them to be perfectly elastic.