Problems

A system of curves is given by the equation \(f(x,y,c) = 0\), where \(c\) is a variable parameter. Show that, in general, the curve obtained by eliminating \(c\) from \(f=0, \partial f/\partial c=0\) touches each curve of the system. \(OX, OY\) are axes of coordinates, not necessarily rectangular. From a point \(P\) on a plane curve lines parallel to \(OY, OX\) are drawn to cut \(OX, OY\) at \(M, N\). \(Q\) is the point of contact of \(MN\) with its envelope, and the tangent at \(P\) to its locus meets \(MN\) at \(R\). Prove that \(OQ, OR\) are harmonic conjugates with respect to \(OX, OY\).

Prove that, if \(f(x,y)\) is a function of \(x^2+y^2\) only, it satisfies the identical relation \[ y \frac{\partial f}{\partial x} = x \frac{\partial f}{\partial y}. \] By changing to polar coordinates, or otherwise, prove conversely that, if \(f(x,y)\) satisfies this relation identically, it must be a function of \(x^2+y^2\) only.

Forces of magnitudes 1, 3, 2, \(-2\), \(-1\), \(-3\) act along the sides \(\vec{AB}, \vec{BC}, \vec{CD}, \vec{DE}, \vec{EF}, \vec{FA}\) of a regular hexagon. Find the direction and the line of action of the resultant.

A uniform rod of length \(2l\) and weight \(W\) is hung from a fixed point by two light elastic strings each of natural length \(l\) and modulus of elasticity \(\lambda\). Shew that the tension \(T\) in the strings is given by \[ W^2(T+\lambda)^2 = 4T^3(T+2\lambda). \]

Shew that the resultant \(R\) of concurrent and coplanar forces \(P_1, \dots, P_n\) is given by \[ R^2 = \sum_{i=1}^n P_i^2 + 2 \sum_{i \ne j} P_i P_j \cos \widehat{P_i P_j}, \] where \(\widehat{P_i P_j}\) is the angle between the forces \(P_i, P_j\). \(Q_1, \dots, Q_n\) is another system of forces such that \(Q_i\) acts in the same line as \(P_i\). Shew that, if the resultant of the system \(Q_1, \dots, Q_n\) is perpendicular to the resultant of the system \(P_1, \dots, P_n\), then \[ \sum_{i=1}^n P_i Q_i + \sum_{i \ne j} (P_i Q_j + P_j Q_i) \cos \widehat{P_i P_j} = 0. \]

One end of a string is attached to a fixed point \(O\) and the other end is attached to the end \(A\) of a uniform rod \(AB\). The end \(B\) of the rod lies on a rough horizontal plane. If \(\mu\) is the coefficient of friction between the rod and the plane and \(\alpha, \beta\) are the angles of inclination of the string and the rod to the horizontal, shew that in the equilibrium position \[ \tan\alpha - 2\tan\beta > \frac{1}{\mu}, \] provided that \(\alpha, \beta\) are both acute angles and that \(O\) and \(B\) are on opposite sides of the vertical through \(A\).

A bead moves without friction on a fixed circular wire; it is repelled from a fixed point of the wire by a force \(F\) which depends on the distance \(r\) between the bead and the fixed point. Find \(F\) in terms of \(r\) so that the reaction between the bead and the wire is the same for all positions of the bead.

A light rope hangs over a light pulley. A mass \(M\) is attached to one end of the rope and a man of mass \(M\) clings to the other end of the rope. The man then climbs the rope with a uniform velocity \(v\) relative to the rope. Describe the ensuing motion. Explain how energy and linear momentum are conserved in this case.

A particle of unit mass moves in a straight line \(OX\) and is repelled from the fixed point \(O\) by a force equal to \(kx^2\) times the distance between the particle and \(O\). When the particle is at a distance \(x_0\) from \(O\) it has a velocity \(v_0\) towards \(O\). Describe the motion of the particle in the three cases \(kx_0^2 \lesseqgtr v_0^2\). In the case \(kx_0^2 > v_0^2\) find the least distance of the particle from the origin.

A particle of unit mass is projected vertically upwards with velocity \(v_0\) under gravity and it is acted upon by a resistance \(kv^2\), where \(v\) is the velocity of the particle. Shew that the maximum height of the particle is \(\frac{1}{2k} \log\left(1 + \frac{kv_0^2}{g}\right)\). Find the velocity of the particle when it returns to its initial position.