A four-wheeled truck of weight \(W\) has wheels of radius \(r\); the distance between the axles is \(l\), and the centre of gravity is equidistant from them. It is standing on level ground with the front wheels in contact with a vertical step of height \(h\) (less than \(r\)). Show that the value of the least force which, applied horizontally to the truck at a height \(a\) (less than \(r\)) above the ground, will cause the wheels to begin to mount the step is \[ \frac{Wl}{2(r+l\tan\alpha-a)}, \] where \(\sin\alpha = \frac{r-h}{r}\). Friction at the axles is to be neglected.
A solid sector is cut out from a uniform solid sphere, of radius \(a\), by a cone of semi-angle \(\beta\) whose vertex is at the centre. Show that the mass-centre of this sector is distant \(\frac{3}{8}a(1+\cos\beta)\) from the vertex. If this solid rests in equilibrium with its spherically curved surface in contact with a rough plane which is inclined at an angle \(\beta\) to the horizontal, show that the axis of symmetry is inclined at an angle \(\sin^{-1}(\frac{2}{3}\tan\frac{1}{2}\beta)\) to the vertical.
A uniform circular hoop of weight \(W\) is suspended on a rough horizontal peg, the angle of friction being \(\lambda\). A vertical force is then applied at a certain point on the hoop and gradually increased until the hoop begins to slip on the peg. Show that the least force which will produce slipping in this way is \(\frac{W \sin\lambda}{1+\sin\lambda}\).
Four light rods are hinged together at their ends to form a quadrilateral \(ABCD\). \(AB=a, CD=b, AD=BC\), and when \(AB\) and \(CD\) are parallel the distance between them is \(c\). The rod \(AB\) is held in a vertical position and the others are adjusted to form a trapezium, \(CD\) being then also vertical: the hinges are then tightened so that relative rotation at each of them would be resisted by a friction couple \(M\). Show that the greatest weight which can be supported at C without displacing the rods is \(\frac{2M(a+b)}{bc}\).
A rod of length \(a\) moves so that its ends \(P\) and \(Q\) always lie on two fixed lines \(OA\) and \(OB\) respectively. The angle \(AOB\) is \(120^\circ\). At the instant when \(OP = \frac{a}{\sqrt{7}}\) and \(OQ = \frac{2a}{\sqrt{7}}\), \(P\) is moving with a velocity \(v\) away from \(O\). Determine, graphically or otherwise, the magnitude of the velocity of the middle point of \(PQ\) and the angular velocity of \(PQ\).
A particle of mass \(m\) is set in motion in a straight line on a smooth horizontal plane by a horizontal force which, starting from zero, increases uniformly to a value \(P\) in time \(T\), falls uniformly to \(-P\) in a further interval \(2T\), and thereafter fluctuates uniformly between these values, passing through zero at intervals of \(2T\). Sketch the form of the velocity-time graph for the motion of the particle, and find the distance it will have travelled after a time \(4nT\), where \(n\) is an integer.
A bead, of mass \(m\), is on a fixed smooth horizontal wire in the form of the equiangular spiral \(r=ae^{k\theta}\). The bead is fastened to one end of a light inextensible string which passes through a small smooth fixed ring at the point \(r=0\), and carries a particle of mass \(M\), at its other end. If the system is released from rest with the string taut, so that \(M\) descends vertically, show by the use of the energy equation (or otherwise) that the acceleration of \(M\) is \(Mgk^2/\{m(1+k^2)+Mk^2\}\).
A jet of water, moving at a speed of 64 ft./sec., impinges normally, without appreciable rebound, on a vertical door. If the force exerted on the door is 250 lb. wt., find in sq. in. the cross-sectional area of the jet. If this water is being pumped from a pond whose surface is 20 ft. below the jet, find the horse-power at which the pump is working. [Take \(g\) to be 32 ft./sec.\(^2\), and assume that the mass of one cubic foot of water is 62\textonehalf{} lb. Frictional losses are to be neglected.]
A light rod is freely hinged to a fixed point at one end \(A\) and has a heavy particle attached to the other end \(B\). It is held in a vertical position with \(B\) uppermost by means of an elastic string attached to the end \(B\) and to a point \(C\) vertically above it. The tension in the string is then equal to half the weight of the particle: \(AB=a\), and \(BC=b\), where \(b
A flywheel with radius \(r\) and moment of inertia \(I\) is mounted in smooth bearings with its axle horizontal. The flywheel being at rest, an inelastic particle of mass \(m\), falling vertically with velocity \(v\), strikes the rim at a point where the radius makes an angle \(\alpha\) with the vertical, and adheres without rebound. Determine the angular velocity of the flywheel immediately after the impact, and also when it has turned through an angle \(\frac{3}{2}\pi - \alpha\).