Problems

Prove that a real rational function of \(x\) may be expressed as the sum of a polynomial and real partial fractions. \par Express in this way \(\frac{1}{x(x^2+1)^2}\).

A uniform thin rigid plank of weight \(W\) has one end on rough horizontal ground and rests, at an inclination \(2\alpha\) to the horizontal, against a rough heavy uniform cylinder which is in contact with the ground along a generator parallel to, and at a distance \(d\) from, the lower end of the plank. The vertical plane through the middle line of the plank contains the centre of gravity of the cylinder. The coefficient of friction between the plank and the ground, the cylinder and the ground, and the plank and the cylinder is \(2 \tan\alpha\) in each case. The length of the horizontal projection of the plank is \(c\). Prove that the only condition necessary for equilibrium is \(3c<4d\). \par Prove that a body of weight \(\frac{1}{2}W\) can be placed in any position on the middle line of the plank without equilibrium being broken if \(c

A uniform elliptic cylinder of weight \(W\) is loaded with a particle of weight \(kW\) at an end of the major axis of the normal cross-section through its centre of gravity, and is placed with its axis horizontal on a smooth horizontal table. Determine the possible positions of equilibrium and consider the stability of the symmetrical positions.

A particle is projected with velocity \(V_0\) at an angle \(\alpha\) with the horizontal and moves under the action of gravity and of an air resistance, equal to \(f\) per unit mass, which is directly opposed to the direction of motion and whose magnitude is given as a function of the velocity of the particle. After time \(t\) the length of the path described is denoted by \(s\), the resultant velocity by \(V\), its horizontal component by \(u\), and the tangent of the angle between the direction of motion and the horizontal by \(p\). Prove that \[ \frac{du}{ds} = -\frac{u}{V^2}f \] and \[ \frac{dp}{dt} = u \sqrt{1+p^2} \frac{dp}{ds} = -\frac{g}{u}, \] and that in the case when \(f=kV^2\), \(k\) being constant, \[ u = V_0 e^{-ks} \cos \alpha \] and \[ F(p_0) - F(p) = \frac{g}{kV_0^2 \cos^2\alpha} (e^{2ks} - 1), \] where \(p_0 = \tan\alpha\) and \[ F(p) = p\sqrt{1+p^2} + \log(p+\sqrt{1+p^2}). \] Deduce that \[ t = \frac{1}{\sqrt(kg)} \int_p^{p_0} \frac{dp}{\sqrt{G(p)}}, \] where \[ G(p) = F(p_0) - F(p) + \frac{g}{kV_0^2 \cos^2\alpha}. \]

A light rigid rod of length \(l\), carrying a heavy particle rigidly attached at one end, is whirled with constant angular velocity \(\omega\) round the vertical through the other end, which is fixed. If \(\omega^2 > g/l\), shew that it is possible for the particle to move steadily in a circle, and that its distance from the vertical axis of rotation is then \(\sqrt{l^2 - g^2/\omega^4}\). \par If, with the angular velocity \(\omega\) maintained unaltered and constant, the distance of the particle from the axis is very slightly altered from the value \(\sqrt{l^2 - g^2/\omega^4}\), find the period of the resulting small oscillations.

(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 \[ \sin\phi\cos(\theta+2\phi)(\mu+\tan\phi) = \frac{W_2 r \sin\theta}{W_1 a}. \]