Problems

If the medians from \(B\) and \(C\) of a triangle \(ABC\) are inclined at an angle \(\frac{1}{3}\pi\), then \[7a^4 + b^4+c^4 = 4a^2 (b^2 + c^2) + b^2c^2.\]

Prove that \[\cos^2\alpha\sin(\beta-\gamma) + \cos^2\beta\sin(\gamma-\alpha) + \cos^2\gamma\sin(\alpha-\beta)\] vanishes with \(\cos(\alpha+\beta+\gamma)\).

\(OAB\) is a vertical circle of radius \(a\). \(O\) is its highest point; \(OA\) subtends angle \(\alpha\) at the centre; \(AB\) subtends angle \(2\beta\). \((\alpha+\beta < \frac{1}{2}\pi.)\) Shew that the time taken for a particle to slide down the chord \(AB\) from rest at \(A\) is \(2\sqrt{(a\cos\alpha/g)}\), when the angle of friction is also \(\alpha\). Shew that if the motion is also subject to a resistance proportional to the velocity, the time of descent is still independent of \(\beta\).

Shew that a system of coplanar forces can in general be reduced (i) to a single force acting at an arbitrarily chosen point in the plane together with a couple; (ii) to \(n\) forces acting along the sides of an arbitrarily chosen polygon of \(n\) sides in the plane. What are the exceptions? \(ABC\) is a triangle, and \(E\) and \(F\) are points on \(BA, BC\). Forces are completely represented by the lines \(EB, BF, FA, AC, CE, AB, BC\). Determine the resultant force in magnitude and direction, and shew that its distance from the point \(A\) is the sum of the distances of \(E, F\) from a line through \(B\) parallel to \(AC\). Indicate roughly the position of the resultant on a diagram.

\(ABB'\) is a straight line and \(CB=CB'\). Shew that the distance between the centres of the circles inscribed in \(ABC\) and \(AB'C\) is \(\frac{1}{2} BB' \sec \frac{A}{2}\).

Prove that the inscribed circle of the triangle \(ABC\) will pass through the centre of perpendiculars if \[2 \cos A \cos B \cos C = (1-\cos A)(1-\cos B)(1-\cos C).\]

A gun fires a shell with a muzzle velocity 1040 feet per second. Neglecting the resistance of the air, what is the furthest horizontal distance at which an aeroplane at a height of 2500 feet can be hit and what gun elevation is required? Shew that the shell would then take approximately 44.2 seconds to reach the aeroplane. [\(g=32\).]

Enunciate the Principle of Virtual Work and the converse theorem. Prove the theorem and its converse for the case of coplanar forces acting on a rigid body; and explain the application to the solution of problems

1. [(i)] where the external forces are given and a position of equilibrium is to be determined;
2. [(ii)] where the external forces are given and it is required to find the internal stresses in the system in a given position of equilibrium.

A light lever \(AOB\) of length \(2a\) can turn freely about its middle point \(O\). A weight \(2w\) hangs from the end \(A\). A light rod \(BC\) of length \(b\) is smoothly jointed to \(AB\) at \(B\), and the end \(C\) supports a weight \(w\) and is restricted by a frictionless constraint to move in the downward vertical through \(O\). Investigate the positions of limiting equilibrium and their stability according as \(a\) is greater than, equal to, or less than \(b\). Also shew that, if, in the case \(b>a\), turning at the joint \(B\) is resisted by a constant friction couple \(F\), there is a position of limiting equilibrium in which the lever makes an angle \(\theta\) with the vertical given by \[ 4waF \sin\theta \cos^2\theta = (b^2-a^2)(wa\sin\theta - F)^2.\]

Eliminate \(\theta\) and \(\phi\) from \begin{align*} \sin\theta + \sin\phi &= a, \\ \cos\theta + \cos\phi &= b, \\ \sin 2\theta + \sin 2\phi &= 2c. \end{align*} Prove that, if \[\sin\alpha + \sin\beta + \sin\gamma + \sin\alpha\sin\beta\sin\gamma = 0,\] then \[\sec^2\alpha + \sec^2\beta + \sec^2\gamma = 1 \pm 2 \sec\alpha\sec\beta\sec\gamma.\]

The functions \(u, v\) satisfy the equations \[ \Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad \Delta v = 0,\] \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}, \] and \(\alpha\) is any real constant. Show that for any pair of values \((x, y)\) for which \(u, v\) are not both zero \[ \Delta (u^2+v^2)^\alpha = 4\alpha^2 \left\{ \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2 \right\} (u^2+v^2)^{\alpha-1}.\]