(i) Define the principal axes of inertia of a plane lamina. \par Find the moment of inertia of a plane lamina about a line in its plane at a distance \(h\) from its centre of mass and inclined at an angle \(\alpha\) to a principal axis of inertia about which its moment of inertia is \(A\), the moment of inertia about the other principal axis being \(B\). \par (ii) When a rigid body of mass \(M\) oscillates about a fixed horizontal axis, shew that the length of the equivalent simple pendulum is \(I/(Mh)\), where \(I\) is its moment of inertia about the axis and \(h\) the distance of its centre of gravity from the axis. \par (iii) A uniform plane rectangular lamina \(ABCD\), with sides \(AB\) of length 8 inches and \(AD\) of length 6 inches, oscillates about a horizontal axis which is parallel to the diagonal \(BD\) and intersects \(AD\) at \(E\). Find the values of the length of \(AE\) for which the length of the equivalent simple pendulum is 4 inches.
Four uniform rods, each of length \(2l\) and weight \(W\), are freely jointed together to form a rhombus \(ABCD\). The rods \(AB, AD\) rest on two smooth pegs in the same horizontal line at a distance \(2a\) apart; \(A\) is uppermost, and the rods \(AB, AD\) each make an acute angle \(\theta\) with the downward vertical. The rhombus is kept from collapsing by a light rod connecting \(B\) and \(D\). Show that the tension in the rod is \[ W \left[ \frac{a}{l \sin^2\theta \cos\theta} - 2\tan\theta \right]. \]
A uniform beam \(AE\) of weight \(W\) and length \(8a\) rests symmetrically on two supports \(BD\) which are in the same horizontal line and are at a distance \(4a\) apart. A weight \(W\) is suspended from the beam at the point \(C\) such that \(BC=a, CD=3a\). Find the shearing force and bending moment at all points of the beam, and show that the maximum bending moment is \(\frac{11}{4}aW\).
A uniform chain of weight \(w\) per unit length hangs from two points at the same level and at a fixed distance \(2l\) apart. If the tension at the ends of the chain is as small as possible, show that the length of the chain is \(\frac{2l}{\theta}\sinh\theta\), where \(\theta\) is given by \(\tanh\theta = 1/\theta\).
A uniform plank of weight \(W_1\) and length \(2a\) is attached by a smooth horizontal hinge at its lower end to a plane which is inclined at an angle \(\theta\) to the horizontal. A uniform circular cylinder of weight \(W_2\) and radius \(r\) rests on the plane with its axis horizontal and supports the plank, the cylinder being above the hinge. All the surfaces have the same coefficient of friction \(\mu\). Show that, if the cylinder is about to move downwards, it must roll on the plane, and show that the angle \(2\phi\) between the plane and the plank is given by \[ \sin\phi\cos(\theta+2\phi)(\mu+\tan\phi) = \frac{W_2 r \sin\theta}{W_1 a}. \]
A particle is projected with velocity \(V\) from a point on an inclined plane in such a way that when it reaches the plane again it strikes it at right angles. Show that the range on the plane is \[ \frac{V^2}{g} \frac{2\sin\alpha}{1+3\sin^2\alpha}, \] where \(\alpha\) is the angle which the plane makes with the horizontal.
A train can be accelerated by a force of 55 lb. weight per ton, and, when steam is shut off, can be braked by a force of 440 lb. weight per ton. Find the least time between stopping points 3850 ft. apart; find the greatest velocity attained and the horse-power per ton weight required at this greatest velocity.
A light inelastic string passes over two light pulleys which lie in the same vertical plane. Between the pulleys, the string hangs vertically and passes under a third light pulley carrying a load of mass \(m_3\). The free ends of the string hang vertically and carry masses \(m_1\) and \(m_2\). If the system is released from rest, show that the downward acceleration of the centre of gravity of the three masses is \[ g \left[ 1 - \frac{16m_1 m_2 m_3}{(m_1+m_2+m_3)(4m_1 m_2 + m_2 m_3 + m_3 m_1)} \right]. \]
A uniform smooth spherical ball of mass \(m\) suspended by a light inextensible string from a fixed point hangs at rest under gravity, and a similar ball falls vertically and strikes the first when travelling with velocity \(v\). At the instant of impact the line of centres of the spheres makes an angle \(\alpha\) with the vertical. Prove that the loss of energy at impact is \[ \frac{1}{2}mv^2 \frac{1-e^2}{1+2\tan^2\alpha}, \] where \(e\) is the coefficient of restitution.
Two particles of masses \(2m\) and \(m\) are attached to the ends of a light elastic flexible string of natural length \(a\) and modulus \(mg\), which is free to slip over a smooth peg. Initially the mid-point of the (unstretched) string is held in contact with the peg, and the particles hang in equilibrium on either side of the peg. The mid-point is now set free. If \(x\) and \(y\) denote the depths of the particles below the peg at time \(t\) subsequent to the release, show that \[ 2\ddot{x}-\ddot{y} = g, \quad \ddot{x}+\ddot{y} = \frac{7}{2}g - \frac{3g}{2a}(x+y). \] Verify that \begin{align*} x &= \frac{13a}{9} - \frac{1}{6}gt^2 + \frac{a}{18}\cos nt, \\ y &= \frac{8a}{9} - \frac{1}{6}gt^2 + \frac{a}{9}\cos nt, \end{align*} where \(n^2=3g/(2a)\), these expressions being valid until one particle reaches the peg.