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 \(\omega\).
Owing to wave formation a yacht has a critical speed which cannot be exceeded in ordinary circumstances. This speed is related empirically to the length of the yacht by the approximate formula \[ \text{critical speed in ft./sec.} = k \sqrt{(\text{length in ft.})}, \] where the constant \(k\) has the approximate value 2.5. Show that, if the critical speed is assumed to depend only on the length, the density of water and the acceleration due to gravity (\(g\)), the proportionality between the critical speed and the square root of the length can be predicted by consideration of dimensions. Find the dimensions of the constant \(k\). Find also the formula relating the speed in metres per second to the length in metres.
A heavy uniform rod of length \(2l\) is placed in a vertical plane so that it is partly supported by a rough horizontal peg while its lower end rests against a smooth vertical wall. The axis of the peg is parallel to the wall and is at a distance \(d\) from it, and the co-efficient of friction between the peg and the rod is \(\mu\). The rod is inclined at an angle \(\alpha\) to the wall and \(l \sin \alpha > d\). Show that the rod will not slip, if \[ l \sin^3\alpha (1+\mu \cot \alpha) \ge d, \quad \text{or} \quad l \sin^3\alpha (1-\mu \cot \alpha) \le d \] according as \[ l \sin^3\alpha < d, \quad \text{or} \quad l \sin^3\alpha > d \] respectively.
A uniform rigid beam of weight \(W\) is clamped at one end so that the end is kept horizontal, and the beam is otherwise unsupported. A weight \(W\) is hung from the centre of the beam. Find the bending moment and the shearing stress at all points of the beam.
The moments of a system of forces acting in the \(Oxy\) plane taken about the points \((0,0), (1,0), (0,1)\) are \(\alpha, \beta, \gamma\) respectively. Find the magnitude and line of action of the resultant. Another system of forces give a resultant which has the same magnitude and direction as in the previous case, but its line of action is \(K\) times as far from the origin. If the moments of this system about the same points are \(\alpha', \beta', \gamma'\) respectively, express \(\alpha', \beta', \gamma'\) in terms of \(K\) and \(\alpha, \beta, \gamma\).
The ends of a rigid rod of length \(l\) are constrained to move along two fixed straight rods which are mutually inclined at an angle \(\phi\). Prove that the loci of the instantaneous centre of rotation relative to the moving rod and relative to the fixed rods are circles. Find the radii of these circles.