Give a rough sketch of the curve whose coordinates are given by \[ \begin{cases} x = a\phi+b\sin\phi, \\ y = b(1-\cos\phi), \end{cases} \] where \(\phi\) is a parameter, and \(a>b>0\). Find the equation of the normal at any point and show that the coordinates of the centre of curvature are \begin{align*} x &= a\phi - b\sin\phi - (a^2-b^2)\frac{\sin\phi}{b+a\cos\phi}, \\ y &= 3a+b-\frac{b^2}{a}+b\cos\phi + \frac{(a^2-b^2)^2}{ab}\frac{1}{b+a\cos\phi}. \end{align*}
\(ABC\) is a triangular lamina, and \(D, E, F\) are points in the sides \(BC, CA, AB\) respectively such that \(BD=\frac{1}{3}BC\), \(CE=\frac{1}{3}CA\), \(AF=\frac{1}{3}AB\). Forces of magnitudes \(kAD, kBE, kCF\) act along \(AD, BE, CF\) respectively. Show that these forces are together equivalent to a couple of moment \(k\Delta\), where \(\Delta\) is the area of the triangle \(ABC\).
A straight rod \(ABC\) of weight \(3W\) rests horizontally on a nearly flat surface, making contact only at the points \(A, B\) and \(C\), where \(B\) is the mid-point of \(AC\). The normal reaction at each of the points of contact is \(W\) and the coefficient of friction is \(\mu\). A gradually increasing horizontal force at right angles to the rod is applied to the rod at \(A\). Find the greatest magnitude of this force for which equilibrium is possible, and describe how the equilibrium is broken.
A rough circular cylinder of radius \(r\) is fixed with its axis horizontal. A uniform cubical block of weight \(W\) with edges of length \(a\) is placed symmetrically upon the cylinder with four edges vertical and four edges parallel to the axis of the cylinder. The block is then rolled upon the cylinder, without slipping, until it has turned through an angle \(\theta\). Calculate (i) the work \(V\) done against gravity in this rotation, (ii) the moment \(M\) of the couple needed to hold the block in equilibrium in the position \(\theta\), indicating the sense of this moment. Hence determine the condition that the position of equilibrium \(\theta=0\) should be stable.
To a motorist driving due West along a level road with constant speed \(V\) the wind appears to be blowing in a direction \(\alpha\) East of North. When he is driving with the same speed \(V\) due East, the apparent direction of the wind is \(\beta\) East of North. Show that, when he is driving at a speed \(2V\) due East, the apparent direction of the wind is \[ \tan^{-1} \left(\frac{1}{3} \tan\beta - \tan\alpha\right) \] East of North. Find also the true speed of the wind.
A train of mass \(M\) lb. is pulled along a level track by an engine which works at a constant rate. The resistance to the motion is \(kv^2\) lb.-wt., where \(v\) is the speed in feet per second. If the maximum speed is \(V\) ft./sec., find the horse-power of the engine, and calculate the distance travelled while the speed increases from \(v_1\) to \(v_2\) ft./sec.
Obtain the expressions \(v^2/a\) and \(dv/dt\) for the components of acceleration of a particle moving with variable speed \(v\) in a circle of radius \(a\). A uniform hollow cylinder of mass \(m\) and radius \(a\) can rotate freely about its axis, which is horizontal. A particle of the same mass \(m\) is placed on the inner surface of the cylinder. Show that, if the cylinder makes complete revolutions and the particle does not leave or slide upon the cylinder, the angular velocity \(\omega_0\) of the cylinder when the particle is at its lowest point must not be less than \(\sqrt{(3g/a)}\), and the coefficient of friction between the particle and the cylinder must not be less than \[ \frac{1}{4}g(a\omega_0^2-3g)^{-\frac{1}{2}}(a\omega_0^2+g)^{-\frac{1}{2}}. \]
A projectile is fired in a given vertical plane with given speed from a point on an inclined plane. Prove that, if the range has its maximum value, the direction of projection is at right angles to the direction of flight just before the projectile reaches the inclined plane. Air resistance is to be neglected.
A particle of mass \(m\) moves in a plane under the action of a force whose components referred to rectangular axes are \((-mn^2x, -mn^2y)\), where \((x,y)\) are the co-ordinates of the particle. Prove that the particle moves in an ellipse with period \(2\pi/n\) and that \[ \frac{1}{2}m(\dot{x}^2+\dot{y}^2) + \frac{1}{2}mn^2(x^2+y^2) \] remains constant. Give the interpretation of this equation. If the particle is initially projected from the point \((a,0)\) with the velocity \((u,v)\), show that the axes of the ellipse lie along the bisectors of the co-ordinate axes provided \(v^2-u^2=a^2n^2\).
Two pulley wheels \(A, B\) of radii \(a, b\) and moments of inertia \(I, K\) respectively are mounted on parallel axes, and can rotate without friction in a common plane. A light endless belt passes round their rims. If a constant couple \(M\) is applied to \(A\), find the angular acceleration of \(B\), assuming that the belt does not slip. Find also the angle through which \(B\) turns while its angular velocity increases from \(0\) to \(\omega\).