Problems

A particle moves without friction inside a narrow straight tube which rotates about one end A with constant angular velocity in a vertical plane. The particle is at relative rest at the mid-point of the tube when vertically below A, and it leaves the tube when the latter is next inclined at 60\(^\circ\) to the downward vertical. Taking \(\cosh\pi/3 = \frac{7}{5}, \sinh\pi/3 = \frac{6}{5}\), shew that the initial direction of motion of the particle on leaving the tube is inclined to the horizontal at an angle 14\(^\circ\) 43' approximately.

Defining a cycloid as the path traced out by a marked point on the circumference of a circle which rolls along a straight line, shew that its parametric equations may be written in the form \[ x=a(\theta+\sin\theta), \quad y=a(1-\cos\theta). \] A smooth groove of this form is fixed in a vertical plane with Oy vertically downwards. A particle is projected from the origin and moves in the groove, which is on the upper side of the cycloid. If the particle leaves the groove when \(y=\frac{1}{2}a\), find the speed of projection.

A particle of mass \(m_0\) is projected with speed \(v_0\) along an upward line of greatest slope of a smooth plane inclined to the horizontal at an angle \(\alpha\). The particle picks up matter, previously at rest, in such a manner that its mass increases with distance traversed at a uniform rate \(k\). Shew that when the particle is instantaneously at rest its mass is \[ m_0\left(1+\frac{3kv_0^2}{2m_0 g \sin\alpha}\right)^{1/3}. \]

Find the moment of inertia of a uniform rectangular lamina about a diagonal in terms of the mass and the lengths of the sides. \par A uniform rectangular lamina ABCD of mass M is free to rotate about the diagonal BD, which is horizontal, and a particle of mass \(m\) is attached to the lamina at C. When the system is in stable equilibrium an impulse is applied at its mass centre, and perpendicular to the lamina. If the lamina is instantaneously at rest when horizontal, determine the magnitude of the impulse.

Solve the simultaneous equations \begin{align*} x + y + \lambda z &= \mu, \\ 2x + 3y + 4z &= 0, \\ 3x + 4y + 5z &= 1 \end{align*} for general values of \(\lambda, \mu\). Examine the cases arising from particular values of \(\lambda, \mu\) for which the equations do not have (i) a unique solution, (ii) a finite solution.

By means of a graph, or otherwise, determine the values of \(\lambda\) for which the equation \[ (x-1)^2(x-a) + \lambda = 0 \] has three real roots, where \(a\) is a given constant greater than unity. \par [If any general formula is quoted, it must be proved.] \par Prove that, whatever the values of \(a, \lambda\), the roots \(\alpha, \beta, \gamma\) are connected by the relation \[ \beta\gamma + \gamma\alpha + \alpha\beta - 2(\alpha+\beta+\gamma) + 3 = 0. \]

(i) Solve the equation \[ x^4 - x^3 + x^2 - x + 1 = 0. \] (ii) Find, in terms of \(p\) and \(q\), the cubic equation such that, if \(x\) is any one of its roots, \(px+q\) is also a root.

Express \(\tan n\theta\) in terms of \(\tan\theta\), where \(n\) is a positive integer, and prove your result. \par Evaluate \(\tan\dfrac{\pi}{12}\).

Prove that \[ 1+\cos\theta+\cos 2\theta + \dots + \cos(n-1)\theta = \sin\frac{n\theta}{2} \cos\frac{(n-1)\theta}{2} \operatorname{cosec}\frac{\theta}{2}. \] Examine carefully whether the result remains true as \(\theta \to 0\), and enunciate precisely any theorem on limits to which you appeal.

Sketch the curves \[ \text{(i) } y = x^2-x^3; \quad \text{(ii) } y^2 = x^2-x^3, \] and find their radii of curvature at the origin.