A uniform solid consists of a cone and a hemisphere fastened together so that their plane faces coincide, the diameter of the hemisphere being equal to that of the base of the cone. Show that if the semi-vertical angle of the cone is greater than \(30^\circ\) the solid will always move to an upright position if placed with the surface of the hemisphere on a horizontal plane.
A boat is anchored to the bed of a river by a heavy uniform chain of length \(l\) and weight \(W\). If the wind exerts a horizontal force \(\frac{1}{2}W\) on the boat, and the depth of the river is \(\frac{1}{2}l\), show that one-third of the chain must be lying on the bed of the river. [Hydrostatic pressure on the chain may be ignored.]
Explain the principle of virtual work for a mechanical system in equilibrium, and describe how the principle may be used to determine the internal constraints of the system. A uniform rod is cut into eight parts which are freely jointed to form a regular hexagon \(ABCDEF\) with two struts \(BF\) and \(CE\). If the system is suspended from \(A\), show that the ratio of the thrusts in the two struts is \(5+2\sqrt{3}\).
A rocket of mass \(M\) carries a missile of mass \(m\). The missile is fired in the direction of motion by an explosive charge, which provides additional kinetic energy \(E\) to the rocket and missile jointly. If the speed of the rocket before the firing was \(V\), calculate its speed after firing, and also find the speed of the missile.
A uniform rod of length \(l\) and weight \(W\) is hinged to a fixed point at one end \(A\), and an elastic string of natural length \(l\) and modulus \(W\) is tied to its other end. To the free end of this string is attached a light ring which can slide on a smooth horizontal bar at a height \(2l\) above \(A\). Show that the period of small oscillations about a position of stable equilibrium is \(\frac{2}{3}\pi\sqrt{(l/g)}\).
A uniform circular hoop of mass \(M\) has a particle of mass \(m\) attached to a point on its circumference. The hoop hangs over a rough peg (coefficient of friction \(\mu\)). Find the greatest possible angle which the radius to the particle can make with the vertical.
Calculate the moment of inertia of a uniform solid sphere, of radius \(a\) and mass \(M\), about a tangent. A uniform solid sphere of radius \(a\) and mass \(M\) hangs in equilibrium supported by two vertical strings attached at opposite ends of a diameter of the sphere. If one string is cut find the tension in the other string immediately afterwards.
A uniform rod of length \(2a\) and mass \(m\) is moving with velocity \(v\) at right angles to its length on a smooth horizontal plane when it strikes a small elastic ball of equal mass (coefficient of restitution \(e\)) initially at rest on the plane. If the point of impact is distant \(c\) from the centre of the rod calculate (a) the angular velocity of the rod after impact, and (b) the velocity of the ball.
A heavy uniform beam of length \(2l\) rests on two supports at the same horizontal level and equidistant from the ends. Find the distance between the supports which would make the greatest value of the bending moment as small as possible.
A uniform rod of length \(l\) and mass \(m\) swings in a plane under gravity about one end where it is freely hinged. Given that the maximum deflection from the vertical is \(\alpha\), obtain (a) the angular velocity, and (b) the horizontal and vertical reactions at the hinge, when the rod makes an angle \(\theta\) with the vertical.