Shew that a system of coplanar forces can be uniquely reduced to three forces acting along the sides of an arbitrarily chosen triangle situated in the plane of the forces. Forces of magnitudes 1, 4, 2, 2, 6, 4 act along the sides \(AB, CB, CD, ED, FE, FA\) of a regular hexagon. Find their resultant and replace them by three forces acting along the sides of the triangle formed by \(AB, CD, EF\).
A long ladder of negligible weight rests with one end on a smooth horizontal plane and with the other projecting over the top of a smooth wall of height \(h\). A light inextensible cord is fastened at one end to the ladder and at the other to the foot of the wall, so that the ladder and cord make the same angle \(\theta (< \frac{1}{4}\pi)\) with the horizontal. The ladder and the cord are in a vertical plane at right angles to the wall. Shew that, if a man ascends the ladder, he will be able to reach a distance \[ h(\operatorname{cosec}\theta + \tfrac{1}{2}\tan 2\theta \sec\theta) \] along the ladder. If the plane were perfectly rough and the cord were removed, shew that he would only be able to ascend the smaller distance \[ h(\operatorname{cosec}\theta + \tan\theta\sec\theta). \]
Find the centre of gravity of a uniform solid hemisphere. A solid consists of a hemisphere of radius \(a\) from which a sphere of radius \(\frac{1}{2}a\) has been removed. It rests with its base on a rough plane inclined at an angle \(\alpha\) to the horizontal, and a gradually increasing force is applied at the pole of the hemisphere in a direction up the plane parallel to a line of greatest slope. Shew that the solid slips or tilts according as \(3\tan\lambda+2\tan\alpha\) is less than or greater than 3, where \(\lambda\) is the angle of friction. Find the value of the force required to destroy the equilibrium.
A framework consists of six equal rods freely jointed together to form a regular hexagon \(ABCDEF\), together with two struts \(FB, EC\). All the rods and struts have the same weight \(W\), and the framework is suspended from \(A\). Shew that the forces acting on the ends of the struts \(FB, EC\) are \(W\sqrt{37}\) and \(W\) respectively.
A gun is placed on a hillside which is in the form of a plane inclined at an angle \(\alpha\) to the horizontal. If the maximum muzzle velocity is \(V\), shew that the area within range is \[ \frac{\pi V^4}{g^2 \cos^2\alpha}. \]
Shew that Newton's experimental law connecting the relative velocity of two bodies before and after impact is equivalent to the statement that the ratio of the impulse of restitution to the impulse of compression is equal to the coefficient of restitution. Two equal smooth spheres are in contact on a smooth table. An impulse is applied to one sphere in a horizontal line through its centre. Shew that the maximum angle between the direction of motion of this sphere and the direction of the impulse is \[ \tan^{-1}\left(\frac{1+e}{2\sqrt{2}(1-e)}\right), \] where \(e\) is the coefficient of restitution. It is assumed that the initial impulse has ceased before the impulse of restitution comes into play.
A particle moves in a straight line, the relation between time and distance being \[ t = ax + bx^2, \] where \(a\) and \(b\) are constants. Determine the relations between (i) distance and acceleration, (ii) distance and velocity, (iii) time and velocity. If the particle travels 2000 feet in 1.9 seconds, and its velocity is then 1000 feet per second, shew that its initial velocity was about 1111 feet per second.
A smooth cylinder of radius \(a\) is rigidly fixed along one of its generators to a horizontal plane. A particle which is initially at rest on the top of the cylinder is slightly disturbed and allowed to slide down. Shew that the particle strikes the plane at a distance \[ \frac{5}{27}(\sqrt{5}+4\sqrt{2})a \] from the line of contact of the cylinder and the plane.
The bob of a simple pendulum is executing small oscillations, and when it is 1 cm from its equilibrium position its velocity is 1 cm. per second. If the length of the pendulum is 1 metre and the mass of the bob is 10 grams, shew that the maximum rate at which work is done during the motion is approximately 170 ergs per second.
A uniform rod of length \(2a\) is held at an angle of \(\frac{1}{3}\pi\) to the vertical and dropped from rest without rotation. After it has fallen a distance \(\frac{1}{2}a\) the upper end of the rod is suddenly fixed. Shew that, when the rod becomes vertical, its angular velocity is \[ \frac{5}{8}\sqrt{\frac{3g}{a}}. \]