Problems

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.

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 \[ \frac{(lx+my)^2-n^2}{l^2+m^2} = \frac{(l'x+m'y)^2-n'^2}{l'^2+m'^2}. \]

Evaluate the indefinite integrals \[ \int \frac{dx}{x(x^4-1)^2}, \quad \int xe^x\sin x dx, \quad \int \frac{dx}{(x-1)\sqrt{x^2-1}}. \]

\(AB\) 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 \(\frac{7}{18}\rho l\sqrt{35gl}\).

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.

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.

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 \((0, \frac{1}{2})\).

Two uniform rods \(AB, BC\) of the same material but of unequal lengths are rigidly jointed at right angles at \(B\), and rest inside a smooth sphere; the diameter of the sphere is equal to the length \(AC\). Show that in a position of equilibrium the plane of the rods is vertical, and find the inclinations of the rods to the vertical.

A rod \(AB\) of length \(L\) is suspended from two points on the same horizontal level by two vertical strings of natural length \(l\) and modulus of elasticity \(\lambda\) attached to the ends. If the line density of the rod increases uniformly from \(\rho\) at \(A\) to \(\rho'\) at \(B\), show that the rod is inclined to the horizontal at an angle \(\sin^{-1}\left\{\frac{lg(\rho'-\rho)}{6\lambda}\right\}\).

Prove the formulae \(s=c\tan\psi\), \(y=c\sec\psi\) for a catenary. A heavy string has one end attached to a small heavy body near the edge of a rough horizontal table, the coefficient of friction being \(\mu\). The string passes over a smooth peg which is fixed at the same horizontal level as the table, and then hangs vertically. If the body is on the point of motion, show that the angle \(\alpha\) which the string makes with the horizontal at the body or the peg is given by \[ \frac{\cos\alpha - \mu\sin\alpha}{1+2\sin\alpha} = \mu\sigma, \] where \(\sigma\) is the ratio of the weight of the body to the weight of the string.