The sides of a triangle \(ABC\) are each divided in the same ratio \(\frac{1}{\lambda}\) at the points \(L, M, N\), so that \[ \frac{AN}{NB} = \frac{BL}{LC} = \frac{CM}{MA} = \lambda. \] Forces of magnitude \(\mu BC, \mu CA, \mu AB\) act through \(L, M, N\) respectively, outwards and perpendicular to the sides. Shew that the resultant of the forces is a couple, and that if the sides of the triangle are divided in the ratio \(\frac{1}{\lambda}\) the resultant so obtained is an equal but opposite couple. Find the magnitude of the couple.
Shew that if a light inextensible string be held in contact in a plane with a rough curved contour, the tension \(T\) of the string, the reaction \(R\) per unit length normal to the contour, and the friction \(F\) per unit length, are given by \[ \frac{dT}{ds} = F, \quad T\frac{d\psi}{ds} = R, \] where \(s\) is the length of arc measured from a fixed point and \(\psi\) is the inclination of the tangent. Prove that the greatest possible ratio of the tensions \(T_1, T_2\) at the ends of the string without slipping is given by \(\frac{T_1}{T_2}=e^{\mu\alpha}\), where \(\alpha\) is the total angle (in radians) through which the tangent to the curve turns while in contact with the string.
A structure of light rigid rods freely jointed at \(A, B, C, D, E\), with all angles either \(90^\circ\) or \(45^\circ\), is subjected to vertical loads \(P, Q\) at \(A, B\) respectively, and is supported at \(C\) and \(D\). \centerline{\includegraphics[width=0.7\textwidth]{truss_diagram.png}} Diagram of a roof truss. CDE is a horizontal base, with supports at C and D. ACE and BDE are isosceles right triangles, with right angles at C and D. A and B are connected by a horizontal rod. The angle at E is 90 degrees. Vertical loads Q and P act downwards at A and B. Sketch the force diagram and shew that the rod \(AB\) is in thrust with a force equal to the mean of \(P\) and \(Q\).
A cotton reel has axle with radius \(a\) and flange radius \(b\), and rests on a rough horizontal table (\(\mu\)). The cotton is pulled steadily with force \(P\) in a direction perpendicular to the axis of the reel and inclined \(\alpha\) to the horizontal. \centerline{\includegraphics[width=0.4\textwidth]{cotton_reel.png}} Diagram of a cotton reel on a surface. The force P pulls the string from the top of the axle at an angle alpha to the horizontal. Prove that if \(b\cos\alpha > a\), the reel will tend to roll up the cotton, and will unwind if \(b\cos\alpha < a\). What happens if \(b\cos\alpha=a\)? Shew that the reel can be kept in equilibrium on a rough slope of inclination \(\theta\) merely by holding the thread taut and parallel to the line of greatest slope, provided \[ \mu > \frac{a}{b-a}\tan\theta. \]
A particle is projected from a given point \(O\) with velocity \(U\). Shew that in subsequent motion under gravity the path is a parabola. Find the position of its focus relative to \(O\) in terms of \(U\) and the direction of projection. Prove that for all directions of projection with given velocity \(U\) from \(O\) in a vertical plane, the parabolic paths touch a certain parabola whose vertex is at a distance \(\frac{U^2}{2g}\) vertically above \(O\), and whose focus is \(O\). Hence or otherwise shew that the maximum range up a plane inclined \(\alpha\) to the horizontal is given by that path whose focus lies in the plane.
Obtain the expressions \(\frac{d^2s}{dt^2}, \frac{v^2}{\rho}\) for the tangential and normal components of acceleration of a particle moving in a plane curve. A particle is allowed to slide down a smooth wire bent in the form of a section of a cycloid with vertex downwards. Shew that the oscillation of the particle about the vertex has a period independent of the amplitude and equal to that of a simple pendulum of length \(4a\), where the equation of the cycloid referred to the vertex as origin is given by \[ \begin{cases} x = a(\theta+\sin\theta), \\ y=a(1-\cos\theta), \end{cases} \] where \(\theta\) is a parameter.
Two equal perfectly elastic spheres moving towards each other collide. Shew that their velocities are interchanged. If the two spheres are moving in a plane with equal velocities, and suffer collision at the moment when the direction of their line of centres is inclined at \(\epsilon\) to the direction of their relative velocity, shew that if the angles of inclination between the direction of the velocities of the spheres is \(\alpha\) before collision, and \(\beta\) afterwards, then \(\tan\beta = \cos 2\epsilon \tan\alpha\).
Shew that the kinetic energy of a rigid body moving in a plane with its centre of mass having velocity \(V\), and the body with angular velocity \(\omega\) about it, is \(\frac{1}{2}I\omega^2+\frac{1}{2}MV^2\), where \(M\) is total mass, and \(I\) the moment of inertia about the axis through the centre. A goods train climbs a steady slope of inclination \(\alpha\) with velocity \(V\) feet per second when the last truck becomes uncoupled. Shew that the truck will come to rest instantaneously after travelling a distance given in feet by \(\frac{V^2(I/a^2+M)}{R+Mg\sin\alpha}\), where \(R\) is the steady frictional resistance to the motion of the truck, \(M\) the total mass of the truck, \(a\) the radius of the wheels, and \(I\) the moment of inertia of each of the two pairs of wheels and axle. If the truck is allowed to run down the slope, shew that its velocity on reaching the point at which it was uncoupled is \(V\sqrt{\frac{Mg\sin\alpha-R}{Mg\sin\alpha+R}}\).
Explain the principle of ``Virtual Work'' and its application to the solution of problems in Statics. Four uniform rigid rods of equal weight \(W\) are freely jointed at \(A, B, C, D\) as shewn. The structure is suspended from \(B\) and is kept in shape by an inextensible light string joining \(B\) to \(D\). Shew that the tension of the string is given by \(T=\sqrt{3}.W\). \centerline{\includegraphics[width=0.6\textwidth]{rod_framework.png}} Diagram of a rhombus-like framework ABCD. B is at the top. A and C are at the same horizontal level. D is at the bottom. The angles are given: angle at B is 45+45=90, angle at A is 45+30=75. Angle at C is not specified but appears symmetric to A. The shape is suspended from B.
A framework consisting of five freely jointed bars forming the sides of a rhombus \(ABCD\) and the diagonal \(AC\) rests on a smooth horizontal plane. The angle \(BAC\) is \(\theta\). Four equal forces \(P\) act inwards horizontally at the middle points of the four sides, perpendicular to them. Prove that the tension in \(AC\) is \(P\cos 2\theta \csc \theta\).