A heavy uniform rod of length \(2l\) is placed in a vertical plane so that it is partly supported by a rough horizontal peg while its lower end rests against a smooth vertical wall. The axis of the peg is parallel to the wall and is at a distance \(d\) from it, and the co-efficient of friction between the peg and the rod is \(\mu\). The rod is inclined at an angle \(\alpha\) to the wall and \(l \sin \alpha > d\). Show that the rod will not slip, if \[ l \sin^3\alpha (1+\mu \cot \alpha) \ge d, \quad \text{or} \quad l \sin^3\alpha (1-\mu \cot \alpha) \le d \] according as \[ l \sin^3\alpha < d, \quad \text{or} \quad l \sin^3\alpha > d \] respectively.
A uniform rigid beam of weight \(W\) is clamped at one end so that the end is kept horizontal, and the beam is otherwise unsupported. A weight \(W\) is hung from the centre of the beam. Find the bending moment and the shearing stress at all points of the beam.
The moments of a system of forces acting in the \(Oxy\) plane taken about the points \((0,0), (1,0), (0,1)\) are \(\alpha, \beta, \gamma\) respectively. Find the magnitude and line of action of the resultant. Another system of forces give a resultant which has the same magnitude and direction as in the previous case, but its line of action is \(K\) times as far from the origin. If the moments of this system about the same points are \(\alpha', \beta', \gamma'\) respectively, express \(\alpha', \beta', \gamma'\) in terms of \(K\) and \(\alpha, \beta, \gamma\).
The ends of a rigid rod of length \(l\) are constrained to move along two fixed straight rods which are mutually inclined at an angle \(\phi\). Prove that the loci of the instantaneous centre of rotation relative to the moving rod and relative to the fixed rods are circles. Find the radii of these circles.
A thin uniform heavy rod \(AB\) is bent into a semicircle of radius \(a\), and is hung by a light inextensible string of length \(l\) which is attached to the ends of the rod and passes over a smooth peg. Derive an equation which gives the values of the inclination of the diameter \(AB\) to the horizontal for which equilibrium is possible.
A smooth wedge weighing 5 lb. has three equal parallel edges and its cross-section perpendicular to these edges is a triangle of sides 3, 4 and 5 in. The wedge rests on a smooth horizontal table with the 5 in. wide face in contact with the table. Particles of mass 4 lb. and 3 lb. which rest in equilibrium on the 4 in. and 3 in. faces respectively are joined by a light string which passes over a small pulley at the top edge of the wedge. Show that when the string is cut the wedge begins to move along the table with an acceleration \(12g/209\).
The ends of a light elastic string of modulus of elasticity \(\lambda\), whose unstretched length is \(2l\), are attached to two fixed points which are separated by a horizontal distance \(2l\). A particle of weight \(w\) is attached to the centre of the string. Verify that if \(\lambda = w/2\) the tension in the string is approximately \(0.57w\) when the system is in equilibrium.
A particle is projected in a vertical plane at an angle \(\beta\) (\(<\pi/2\)) to the upward pointing line of greatest slope of a plane which is inclined at an angle \(\alpha\) to the horizontal. Find the value of \(\beta\) which gives the maximum range along the inclined plane for a given speed of projection, and prove that at the extreme range the particle hits the plane at an inclination \((\alpha + \frac{1}{2}\pi)\) to the plane.
A particle of mass \(m\) is suspended by a light inelastic string of length \(l\) from a point \(A\) which is constrained to move in a horizontal circle of radius \(a\) at a constant speed \(a\omega\). Prove that, if the particle can describe a horizontal circle of radius \(x\) with constant speed, then \(x\) satisfies the equation \[ \omega^4 x^2 \{l^2 - (x-a)^2\} = g^2(x-a)^2. \] If \(\omega, l\) and \(a\) are given, show how to decide which of the four roots of this equation can be an actual value of \(x\).
A heavy flywheel consists of a uniform circular disc of radius \(a\) and mass \(M\) which can rotate about an axle through the centre perpendicular to its plane. This axle can be engaged with a coaxial light cylindrical drum of radius \(b\), around which is wound one end of a light rope. The rope supports a cage of mass \(m\) which hangs freely in a vertical shaft. When the flywheel is running freely with angular velocity \(\omega\) it is suddenly engaged with the drum so that the cage begins to rise. Find the height through which the cage rises (assuming that the rope is long enough to prevent the cage hitting the drum).