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).
Prove that a given force acting in the plane of a triangle is equivalent to three forces acting along the sides of the triangle. Find the magnitudes of the three forces if the lengths of the sides of the triangle are 3, 4, 5, while the given force is of magnitude \(F\) and acts in the line bisecting the side of length 4 at right angles.
A tripod formed of three uniform rods \(OA, OB, OC\), which are of the same weight and of the same length \(2a\) and are freely jointed together at \(O\), rests on a rough horizontal plane so that the feet \(A, B, C\) form an equilateral triangle. If the coefficient of friction is \(\frac{1}{4}\), prove that the least possible height of \(O\) above the plane is approximately \(1.79a\).
A wedge is cut from a uniform solid circular cylinder of radius \(a\) by two planes inclined at an angle \(\alpha\). One plane is perpendicular to the axis of the cylinder, and the line of intersection of the planes touches the surface of the cylinder. Prove that the mass-centre of the wedge is at a perpendicular distance \(\frac{3}{8}a \tan\alpha\) from the circular base of the wedge, and explain why the mass-centre is not near the centre of the base when \(\alpha\) is small.
A uniform bar \(AB\) of length \(l\) and weight \(w\) per unit length is attached to a fixed smooth hinge at \(A\) and is kept horizontal by a light chain of length \(2l\) which joins \(B\) to the point vertically above, and distant \(\sqrt{3}l\) from, \(A\). Find the thrust, shearing force and bending moment at any point of the bar.