An equilateral triangle formed of light rods freely jointed stands on its base \(AB\) which is supported at \(A\) and \(B\). A weight of 1 lb. is suspended from the middle point of \(AC\) and a weight of 2 lbs. from the middle point of \(BC\). Draw a diagram shewing the stresses at all the joints.
A cylindrical hole of radius \(a\) is bored through a body and the body is suspended from a rough horizontal peg passing through the hole. Prove that in equilibrium the inclination to the vertical of the plane through the centre of gravity of the body and the axis of the hole is not greater than \(\sin^{-1}\frac{a\sin\lambda}{d}\), where \(\lambda\) is the angle of friction and \(d\) is the distance between the centre of gravity and the axis of the cylinder.
State the principle of virtual work and prove it in the case of any number of particles rigidly connected together. A rod lies in equilibrium with its ends on two smooth planes inclined at angles \(\alpha, \beta\) to the horizontal, the planes intersecting in a horizontal line. Shew that the inclination of the rod to the horizontal is \[ \tan^{-1}\frac{\sin(\alpha-\beta)}{2\sin\alpha\sin\beta}. \]
A force \(P\) raises a weight \(W\) through a certain height and is then removed, so that the weight comes to rest after a time \(t\) from the beginning of the motion. Shew that \[ P=W/\left(1-\frac{t_0}{t}\right) \] where \(t_0\) is the time the weight would take to fall freely from the highest point to its original position.
If a system of particles is acted on by no forces except mutual reactions between the particles, prove that their centre of gravity is either at rest or moves uniformly in a straight line. Two particles of equal mass lie on a smooth table at \(A\) and \(B\), and are connected by a light inextensible string of length \(AB\). One particle receives an impulse perpendicular to the string. Shew that each particle describes a cycloid.
A particle is fastened to a straight elastic string the ends of which are tied to two fixed points. The natural lengths of the two parts of the string are \(a, b\) and initially their tensions are \(T_1, T_2\) (\(T_2>T_1\)). If the particle is then released, shew that it will move through a space \(\frac{2ab}{a+b}\frac{T_2-T_1}{\lambda}\) before coming to rest, provided the string remains stretched throughout, and find the necessary condition that this may be the case.
Find the equation of the path of a projectile whose velocity and elevation of projection are known. If the velocity of projection is 360 feet per second and the projectile is aimed so as to pass through a straight line on the ground at a nearest distance of 1000 yards from the point of projection, prove that the length of the line which can be hit is about 1814 yards.
Investigate the small oscillations of a simple pendulum and find the time of vibration. Two pendulums of lengths 30 inches and \(30\frac{1}{4}\) inches reach the highest point of their swing at the same time. Shew that they will first be simultaneously in this position again after about \(3\frac{1}{2}\) minutes.