A smooth ball of mass \(m\) hangs at rest on a light inextensible string from a fixed point. A second smooth ball of mass \(m'\) impinges directly on the first so that its velocity \(V\) makes an acute angle \(\alpha\) with the string. The coefficient of restitution between the two balls is \(e\). Shew that the initial velocity of the first ball is \[ \frac{m'(1+e)V\sin\alpha}{m+m'\sin^2\alpha}, \] and that the impulse in the string is \[ \frac{mm'(1+e)V\cos\alpha}{m+m'\sin^2\alpha}. \]
\(a\) is the unstretched length and \(kmg\) the modulus of elasticity of a light extensible string, to one end of which is attached a particle of mass \(m\). The other end \(O\) of the string is fixed to a rough plane inclined to the horizontal at an angle \(\alpha\) which exceeds the angle of friction, \(\lambda\), between the particle and the plane. If the particle is also at \(O\) and is released from rest, find an expression for the distance through which it will slide before its velocity is again zero. Shew further that the particle will remain at rest in this new position unless \(3\tan\lambda\) is less than \[ (k^2\sec^2\alpha+8k\sec\alpha\tan\alpha+4\tan^2\alpha)^{\frac{1}{2}} - k\sec\alpha - \tan\alpha. \]
Evaluate the indefinite integrals \[ \int \frac{dx}{x(x^4-1)^2}, \quad \int xe^x\sin x dx, \quad \int \frac{dx}{(x-1)\sqrt{x^2-1}}. \]
Prove that the locus of foci of conics inscribed in the parallelogram formed by the lines \[ lx+my\pm n=0, \quad l'x+m'y\pm n'=0 \] is the rectangular hyperbola \[ \frac{(lx+my)^2-n^2}{l^2+m^2} = \frac{(l'x+m'y)^2-n'^2}{l'^2+m'^2}. \]
An Atwood's machine consists of a light frictionless pulley carrying a light string at one end of which is carried a mass \(A\) of 19 ounces, and at the other end of which is carried a mass \(B\) of 17 ounces and a rider of 4 ounces. The system is released from rest with the rider at a height of 100 inches above a fixed ring through which \(B\) ultimately passes and on which the rider is removed. Shew that the mass \(B\) comes to rest at a depth of 90 inches below the ring, and that it then ascends, picks up the rider and comes to rest once again at a height 81 inches above the ring.
Obtain expressions for the tangential and normal components of the acceleration of a particle moving in a plane curve. A bead is free to move on a smooth wire in the form of a catenary fixed with its axis vertically upwards. The bead is projected from the lowest point of the catenary and, in the subsequent motion, the ratio of the greatest and least values of the reaction between the bead and the wire is 15. Determine the least value of the reaction in terms of the weight of the bead.
Trace the curve \[ 2xy^2 + 2(x^2-x+2)y - (x^2-5x+2) = 0. \] Prove that at no finite real point of the curve is the tangent parallel to the \(x\)-axis. Find the curvature at the point \((0, \frac{1}{2})\).
A triangle \(ABC\) is circumscribed to a conic \(S_1\). Prove that there exists a conic \(S_2\) (not consisting of two sides of the triangle) which is circumscribed to \(ABC\) and has double contact with \(S_1\). Shew that the pencil \[ S_1+\lambda S_2 = 0, \] contains one conic \(S\) for which \(ABC\) is a self-conjugate triangle, and that \(S_1\), \(S_2\) are reciprocal with respect to \(S\). Shew further that the chord of contact passes through the points in which the tangents to \(S_2\) at \(A\), \(B\), \(C\) meet the opposite sides, and that its pole is the point of intersection of the lines joining \(A\), \(B\), \(C\) to the points of contact of \(S_1\) with the opposite sides.
A motor car weighing 10 cwt. travels at a uniform speed of 25 miles per hour up a hill of uniform gradient. The hill is 800 feet high and one mile long. Find the horsepower exerted by the engine in overcoming gravity. If the engine is actually working at 20 horse-power, find the frictional resistance. If the frictional resistance varies as \(v^2\), where \(v\) is the speed, find how far the car would travel on the flat if the engine were disconnected at a speed of 25 miles per hour.
\(AB\) is a straight rod of length \(l\) whose density varies uniformly from \(\rho\) at \(A\) to \(2\rho\) at \(B\). The rod is free to swing about the end \(A\) and is hanging at rest when it receives a horizontal blow through its centre of mass. If the rod next comes to rest when horizontal, shew that the magnitude of the impulse is \(\frac{7}{18}\rho l\sqrt{35gl}\).