Three particles A, B, C of the same mass rest on a smooth horizontal table. AB and BC are taut inextensible strings, and the angle ABC is acute and equal to \(\alpha\). A is set in motion with velocity V parallel to CB. Show that when the string again tightens C starts off with velocity \(\dfrac{V}{3+4\tan^2\alpha}\).
``The principle of virtual work epitomizes the laws of statics.'' State and prove this principle, and discuss the foregoing statement. Illustrate the use of the principle by solving the following problem: \par Two uniform rods AB, BC, of equal lengths and of weights \(W_1, W_2\) respectively, are smoothly hinged together at B and to fixed supports at the same level at A, C, so that the angles CAB and ACB are each equal to \(\beta\). Find the horizontal and vertical components of the action at B.
Establish the formula \[ \frac{1}{2} \int \left(x \frac{dy}{dt} - y \frac{dx}{dt}\right) dt \] for the area of a closed curve given in parametric form \(x=f(t), y=g(t)\), explaining any conventions of sign involved. \par Show that in the parametric representations \[ x = a \cos t, \quad y = b \sin t, \] and \[ x = a \cosh t, \quad y = b \sinh t, \] of the ellipse and the hyperbola, respectively, the area swept out by a radius vector \(OP\) from the centre \(O\), as \(P\) describes an arc of the curve, is proportional to the change in \(t\) along the arc.
A conic circumscribes the triangle ABC and the tangents to it at A, B, C form a triangle PQR. Prove that AP, BQ, CR are concurrent. \par If the conic is a parabola, prove that the point of concurrence lies on the conic touching the sides of the triangle ABC at their middle points.
A body is free to rotate about a fixed axis. Prove that the rate of change of moment of momentum about the axis is equal to the moment of the external forces about this axis. \par A uniform equilateral triangular lamina of side \(2a\) can rotate freely about a fixed horizontal axis coinciding with one side. Find the length of the equivalent simple pendulum.
Two light spiral springs, OA, AB, are joined together at A, and particles of equal mass are fastened to the compound spring at A, B respectively; the end O is fixed at a point of a smooth horizontal table. Throughout the movement of the system O, A, B remain in a fixed horizontal straight line, with A between O and B. If the masses oscillate so that the displacements of A, B along OAB at any instant are \(x_1, x_2\) respectively, obtain the equations of motion \begin{align*} \ddot{x_1} + q^2x_1 - p^2x_2 &= 0 \\ \ddot{x_2} - p^2x_1 + p^2x_2 &= 0, \end{align*} where \(2\pi/p\) is the period of oscillation of the mass B when A is held fixed, and \(2\pi/q\) is the period of oscillation of the mass A when B is held fixed with OB equal to the unstretched length of the combined spring. \par For the case in which \(q^2 = \frac{5}{2}p^2\) show that a solution of these equations can be obtained in which \(x_1 = H \cos \sqrt{\frac{1}{3}} pt\), \(x_2 = K \cos \sqrt{\frac{1}{3}} pt\), and find the ratio \(H/K\). \par Show also that a second solution \(x_1 = H' \cos \sqrt{\frac{5}{3}} pt\), \(x_2 = K' \cos \sqrt{\frac{5}{3}} pt\) exists.
Evaluate \[ \int_0^1 \sqrt{\frac{1-x}{1+x}} dx, \quad \int_0^{\pi/4} \frac{x}{\cos^4 x} dx, \quad \int_0^\infty \frac{x\,dx}{x^3+1}. \]
Spheres are described to touch two fixed planes and to pass through a fixed point. Prove that they all pass through a second fixed point and that the locus of their points of contact with either plane is a circle.
If a particle is describing a circle of radius \(a\) with constant speed \(v\), show that the acceleration is along the radius, and that its magnitude is \(v^2/a\). \par A cylindrical shaft is rotating about its axis, which is vertical, with constant angular velocity \(\omega\), and AOA' is a diameter of the shaft. Light rods AB, A'B' are freely pivoted to the shaft at A, A', and carry at their ends blocks B, B', each of mass \(m\), which slide against the rough inside surface of a fixed cylindrical drum. The drum surrounds the shaft and is co-axial with it; the plane containing AB, A'B' and O is perpendicular to the axis of the shaft, and the angles ABO, A'B'O are equal to \(\alpha\). Show that the couple exerted on the drum is \[ 2\mu m b^2 \omega^2 \sin\alpha / (\sin\alpha + \mu\cos\alpha), \] where \(b = OB\) and \(\mu\) is the coefficient of friction between the blocks and the drum. \par [Diagram of a rotating shaft with pivoted rods and blocks inside a drum is shown]. \par Investigate whether this result still holds when the shaft is rotating in the opposite direction to that shown in the diagram.
Prove the formulae \(\ddot{r}-r\dot{\theta}^2\), \(r\ddot{\theta}+2\dot{r}\dot{\theta}\) for the radial and transverse components of acceleration in polar coordinates. \par A smooth small ring P is fixed to one edge of a smooth horizontal table. A light thread AB passes through P; to one end A, which is on the table, is fastened a mass \(m_1\), and to the other end B, which hangs below P, is attached a mass \(m_2\). If during the motion of the system \(m_1\) moves directly towards P and \(m_2\) moves in a vertical plane, write down the equations of motion of \(m_1\) and \(m_2\) in terms of \(r\), the distance of \(m_2\) from P, and \(\theta\), the angle that PB makes with the vertical. \par If at \(t=0\), when \(m_1\) is at rest and PB is vertical and of length \(l\), \(m_2\) is projected horizontally with velocity \(V\), find the initial values of \(\dot{r}\) and \(\dot{\theta}\) and the initial tension in the thread.