A locomotive of mass \(m\) tons starts from rest and moves against a constant resistance of \(P\) pounds weight. The driving force decreases uniformly from \(2P\) pounds weight at such a rate that at the end of \(a\) seconds it is equal to \(P\). Find the velocity and the rate of working after \(t\) seconds \((t
Define the angular velocity of a lamina moving in any manner in its plane. On a lamina is traced a circle of radius \(a\) which is made to roll along a straight line with constant angular velocity \(\omega\). Find the velocity and acceleration of any point of the lamina. Describe the path of any such point, distinguishing the cases of a point inside, on, and outside the circle. Shew that the locus of points which are at inflexions of their paths is a circle having as diameter the line joining the centre of the given circle and its point of contact with the line on which it rolls. \(C\) is the centre of two concentric circles \(A, B\), and a line \(CPQ\) meets the circles in \(P, Q\). Tangents \(XPX, YQY\) are drawn to the circles at \(P, Q\). The circle \(A\) rolls along the line \(XX\) carrying the circle \(B\) with it, so that \(C, P, Q\) are always collinear, until the point \(P\) is again on the line \(XX\) and \(Q\) is consequently again on the line \(YY\). The distance between the two positions of \(P\) is equal to the circumference of the circle \(A\). Investigate the fallacy in the assertion that the distance between the two corresponding positions of \(Q\) is equal to the circumference of the circle \(B\).
A family of parabolas have a given point as vertex, and all pass through another given point. Prove that the locus of their foci is a cubic curve.
Find the differential coefficient of \[ \tanh^{-1} \left\{ \frac{axp + b(x+p)+c}{qy} \right\}, \] where \(y^2 = ax^2 + 2bx + c\), \(q^2 = ap^2 + 2bp + c\). Hence, or otherwise, evaluate \[ \int \frac{dx}{(x-p)y}. \]
Two equal flat scale pans are suspended by an inextensible string passing over a smooth pulley so that each remains horizontal. An elastic sphere falls vertically and when its velocity is \(u\) it strikes one of the scale pans and rebounds vertically. Show that the sphere takes the same time to come to rest on the scale pan as it would if the scale pan were fixed.
State and prove any theorems you know relating the velocity and acceleration of the centre of inertia of a system of particles to (1) the motions of the individual particles; (2) the forces acting on the system of particles. A light string passes over a fixed smooth pulley and carries at one end a mass \(6m\), and at the other a smooth pulley of mass \(3m\) over which passes a second light string carrying masses \(2m\) and \(m\) at its ends. Assuming that the system moves from rest obtain expressions for the velocities and accelerations of the movable pulley and the masses. Use this system to verify the principle of the conservation of energy, and explain why it does not illustrate the principle of the conservation of linear momentum. What distribution of masses between the various parts of the system would cause it to do so?
Shew that all chords of an ellipse which subtend a right angle at a given point on the ellipse meet in a point \(P\). Shew also that the locus of \(P\) is a concentric, similar and similarly situated ellipse.
Prove that \[ \left(\frac{d}{dx}\right)^n \tan^{-1}x = P_{n-1}(x)/(x^2+1)^n, \] where \(P_{n-1}\) is a polynomial in \(x\) of degree \(n-1\), and \[ P_{n+1} + 2(n+1)xP_n + n(n+1)(x^2+1)P_{n-1} = 0. \]
Three equal particles \(A, B, C\) of mass \(m\) are placed on a smooth horizontal plane. \(A\) is joined to \(B\) and \(C\) by light threads \(AB, AC\) and the angle \(BAC\) is \(60^\circ\). An impulse \(I\) is applied to \(A\) in the direction \(BA\). Find the initial velocities of the particles and show that \(A\) begins to move in a direction making an angle \(\tan^{-1}\sqrt{3}/7\) with \(BA\).
Discuss the simple harmonic motion of a point moving in (1) a straight line, (2) a curve. Obtain the relations which exist between the period, amplitude, velocity at any point, and the acceleration at unit distance from the centre of the path. Give a discussion of the simple and the cycloidal pendulums with particular reference to the relation between the period and the amplitude of the motion in each case. Prove that if a point move in an arc of a parabola having the vertex as middle point so that the motion of the projection of the point on the axis of the parabola is simple harmonic, then the motion of the projection of the point on the directrix is also simple harmonic and of double the period.