A particle is projected under gravity from a point of an inclined plane in a direction that lies in a vertical plane through a line of greatest slope. Prove that the time taken by the particle to attain its greatest distance from the plane is one-half of the time of flight. \par If the particle can be projected with an assigned initial speed so as to pass through a given point of the plane, show that in general there are two directions of projection. Show also that the product of the times of flight is \(2R/g\), where \(R\) is the range.
Define the angular velocity of a rigid lamina moving in its own plane, and prove that, in general, just one point of the lamina is instantaneously at rest. \par Show that, if two such rotations of angular velocities \(\omega_1, \omega_2\) about points \(P_1, P_2\) respectively are superposed, the resultant motion is a rotation of angular velocity \(\omega_1+\omega_2\) about \(G\), the centre of mass of a mass \(\omega_1\) at \(P_1\) and a mass \(\omega_2\) at \(P_2\). \par State and prove the generalisation of this theorem for the case of any number of superposed rotations.
The perpendicular from the origin \(O\) on to the tangent at a point \(P\) of a plane curve \(C\) is of length \(p\) and is inclined to the axis of \(x\) at an angle \(\alpha\). Show that the coordinates of \(P\) are \[ (p \cos \alpha - p' \sin \alpha, \ p \sin \alpha + p' \cos \alpha), \] where the dash stands for \(d/d\alpha\). Show also that the radius of curvature of \(C\) at \(P\) is \(\pm (p+p'')\). \par If \(C\) is a simple closed curve containing \(O\) in its interior and everywhere convex (i.e. lying entirely to one side of each of its tangents), prove that the perimeter of \(C\) is of length \[ \int_0^{2\pi} p d\alpha, \] and encloses an area \[ \frac{1}{2} \int_0^{2\pi} (p^2 - p'^2) d\alpha. \]
Find the length and direction of the major axis of the ellipse \[ 24x^2 + 8xy + 18y^2 = 1. \] Prove that this ellipse lies entirely inside the ellipse \[ 23x^2 + y^2 = 1. \] % Question 10 is cut off in the image
A small ring slides on a smooth circular wire of radius \(a\) fixed in a vertical plane, and is connected to the highest point of the wire by a light spring of natural length \(a\). Prove that, in any motion in which the spring remains taut, the energy is constant. \par If the ring is in equilibrium at one end of the horizontal diameter, and is then slightly disturbed, prove that it will oscillate with period \(2\pi/n\), where \[ n^2 = \frac{\sqrt{2}+1}{2}\frac{g}{a}. \]
Particles \(P_1, P_2, \dots, P_n\) of the same mass are placed on a smooth horizontal table at the vertices of a regular polygon of \(n\) sides each of length \(a\), and are joined by light inextensible strings \(P_1P_2, P_2P_3, \dots, P_{n-1}P_n\) of length \(a\). (There is no string \(P_n P_1\).) The system is set in motion by a given impulse \(T_1\) applied to \(P_1\) along the line \(P_1P_n\). Denoting the impulses in the strings by \(T_2, T_3, \dots, T_n\) respectively, prove that \[ T_r - 2T_{r+1} \sec \frac{2\pi}{n} + T_{r+2} = 0, \] where \(r=1, 2, \dots, n-2\). (It is helpful to consider the component velocities of \(P_r\) and \(P_{r+1}\) along \(P_rP_{r+1}\).) \par Prove that these equations have a solution of the form \[ T_r = A\alpha^r + B\beta^r, \] where \(A, B, \alpha, \beta\) are constants; and, without carrying out the calculations, explain how to determine these constants.
Evaluate: \[ \int \frac{(x+1)dx}{x\sqrt{(x^2-4)}}, \quad \int_0^\infty \frac{dx}{\cosh^3 x}, \quad \int_0^{\pi/2} \frac{\cos x + 2 \sin x + 2}{(1+2\cos x)^2} dx. \]
Prove that the locus of a point in space which is at the same given distance from each of two intersecting straight lines consists of two ellipses with a common minor axis.
A flywheel in the form of a uniform disc of radius 9 in. and mass 250 lb. can rotate without friction about an axis through its centre perpendicular to the disc; a light cord is wound several times round the axle, which is rough and of radius 1 in., and this cord carries at one end a mass of 50 lb., which hangs freely from the axle. If the other end of the cord is pulled with a tension of 100 lb. wt., find the acceleration with which the 50 lb. mass rises. \par Find also what braking couple must be applied, in addition, if the wheel is brought to rest in 10 revolutions from an angular velocity of 5 revolutions per second. Specify the units in which this couple is evaluated. Take \(g\) to be 32 ft./sec.\(^2\)
A light rod \(AB\) of length \(2a\) can rotate freely about one end \(A\). A particle of mass \(m\) is attached to the other end \(B\), and a particle of mass \(2m\) is attached to the mid-point \(M\). The rod is released from rest in a horizontal position. Find its angular acceleration immediately afterwards, and show that the bending moment at \(M\) is then \(\frac{3}{7}mga\). \par Find the bending moment at \(M\) and the tension in the upper half of the rod when the inclination of the rod to the vertical is \(\theta\).