The resistance to motion of a car weighing 2500 lb. when travelling at \(v\) feet per second is \((16+\frac{v^2}{64})\) lb. wt. It will travel on a level road at a uniform rate of 60 miles per hour. Find the horse-power required and write down the equations to be solved to determine the relation between velocity and distance run after reaching an up-slope of 1 in 20,
Shew that, if a particle is moving in an ellipse, its acceleration perpendicular to the radius vector \(r\) from the centre varies as \(\frac{1}{r}\frac{d^2\phi}{dt^2}\), where \(\phi\) is the eccentric angle. Find the corresponding value of the acceleration along the radius vector. Deduce that, if the particle is moving under a centre of force at the centre, that for must vary as the radius vector. (Note: there appears to be a missing part of the question text here).
(i) Shew that a system of forces in one plane can be reduced to either of the following systems, (a) a force acting through an assigned point together with a couple, (b) three forces acting along the sides of a given triangle. (ii) A system of forces is represented in magnitude, direction and line of action by the sides of a convex polygon. Shew that the forces are equivalent to a couple equal to twice the area of the polygon.
Explain the advantages of employing the principle of virtual work in the solution of statical problems. A heavy uniform rod \(AB\) of weight \(W\) and length \(2a\) is free to move in a vertical plane about a hinge at \(A\), and the other end is freely jointed to a similar rod \(BC\). A smooth guide compels \(C\) to move along a horizontal straight line through \(A\). The mid-points of the rods are joined by a light inextensible string of length \(\sqrt{3}a\), and a particle of small weight \(w\) is connected to \(B\) by a second string which passes through a smooth ring at a height \(4a\) vertically above \(A\). Shew that, when the system is in equilibrium with the point \(B\) above \(AC\), the tension in the string joining the rods is \[ \sqrt{3} W - 2w. \]
Five light rods are freely jointed together to form a rectangle \(ABCD\) and its diagonal \(AC\), where the lengths \(AB\) and \(BC\) are 8 and 6 units respectively. Two further light rods \(BE, DE\) are attached to the rectangle at \(B\) and \(D\), and are jointed together at \(E\), where \(E\) is the point on \(AC\) produced, such that \(AC=CE\). The whole framework is suspended by a string attached to \(A\), and weights \(W, W\) and \(2W\) are attached to \(B, E\) and \(D\) respectively. Determine approximately the stresses in the rods.
A heavy uniform rod \(AB\) of length \(6a\) and weight \(W\) is supported by two parallel horizontal bars. The points of contact are \(C\) and \(D\) as shown in the figure, \(AC\) and \(CD\) being each equal to \(a\), and the vertical plane through the rod being perpendicular to the bars. \centerline{\includegraphics[width=0.7\textwidth]{rod_on_bars.png}} The diagram shows a rod AB resting on two bars at C and D. A is the lower end, B the upper end. C is below D. The upper bar \(D\) is rough with coefficient of friction \(\mu\), greater than \(\frac{1}{2}\tan\alpha\), where \(\alpha\) is the inclination of the rod to the horizontal, while the lower bar \(C\) is smooth. A force \(P\) acting vertically downwards is applied at \(A\) and gradually increased. Shew that for a certain range of \(\alpha\) equilibrium is broken by slipping when \[ P = \frac{2\mu - \tan\alpha}{\mu+\tan\alpha} W, \] and find the range of \(\mu\) for this to be possible.
A particle is projected from a point on an inclined plane and moves under gravity so as to strike the plane again at right angles. Shew that the minimum value of the angle between the direction of projection and the horizontal, for varying inclinations of the plane, is \[ \tan^{-1}(2\sqrt{2}). \]
Two masses \(M+m\) and \(M\) are connected by a light inextensible string which passes over a light pulley which is free to turn about a horizontal axis. A rider of mass \(2m\) is placed on the smaller mass and the system is released from rest. When the velocity has attained the value \(V\) the rider is removed by a fixed inelastic ring which allows the mass \(M\) to pass through. The system continues to oscillate, the rider being picked up and deposited alternately. Shew that the system will finally come to rest after a time \[ \frac{(2M+3m)(4M+3m)}{m^2}\frac{V}{g}. \]
A railway train of mass 300 tons has a driving force of \((9-\frac{v}{20})\) tons weight, where \(v\) is the velocity in miles per hour. The resistance to motion is \((a+bv^2)\) tons weight, where \(a\) and \(b\) are constants. The maximum velocity on the level is 60 miles per hour, while on a gradient of 1 in 100 the maximum velocity is 40 miles per hour. The train is travelling at full speed on the level and starts to ascend a gradient of 1 in 50. Shew that the speed is reduced to 30 miles per hour in approximately 1.44 miles. \[ [\log_{10} e = 0.4343.] \]
A smooth uniform sphere rests on a horizontal plane and a second similar sphere is dropped vertically so as to strike the first with velocity \(V\). Shew that the maximum velocity that can be imparted to the sphere which was initially at rest is \((1+e)V/2\sqrt{2}\), where \(e\) is the coefficient of restitution between the two spheres.