Problems

Given an ellipse \(a^2y^2 + b^2x^2 - a^2b^2 = 0\), denote by \(N\) the length of the part of the normal at a point \((x, y)\) between the curve and the \(X\)-axis. Prove that the radius of curvature \(\rho\) at the same point is given by the formula \[ \rho = \frac{N^3 a^2}{b^4}. \]

A uniform rod free to turn about its centre \(O\) rests in a horizontal position. A smooth uniform sphere of mass \(m\) falling vertically strikes the rod directly at a distance \(a\) from \(O\). Prove that the rod turns about \(O\) with angular velocity \[ mua(1+e)/(I+ma^2), \] where \(I\) denotes the moment of inertia of the rod about a perpendicular through \(O\), \(e\) the coefficient of restitution between the sphere and the rod, and \(u\) the velocity of the sphere just before impact. Find also the loss of kinetic energy resulting from the impact.

An elastic string of natural length \(2c\) has its ends attached to the upper corners of a square picture frame of side \(2c\). The string passes over a rough peg and the frame hangs symmetrically below, each half of the string making an angle \(60^\circ\) with the horizontal. The frame is pulled down through a small distance and then released. Shew that it will oscillate up and down and that the period of the small oscillations is the same as that of a simple pendulum of length \(\displaystyle\frac{4c\sqrt{3}}{7}\).

If \(S=0\) and \(p=0\) are the equations of a fixed conic and a fixed line, interpret the equation \[ S + \lambda p^2 = 0, \] where \(\lambda\) is a parameter. Each member of a family of conics touches two fixed lines at fixed points \(A\) and \(B\). Shew that the sides of a given triangle meet the polars of the opposite vertices with respect to any one conic of the family in three points which lie on a straight line, and further that the envelope of this line for all members of the family is a conic which touches the sides of the triangle and also touches the line \(AB\).

Shew that, if \(z = x\phi\left(\frac{y}{x}\right) + \psi\left(\frac{y}{x}\right)\), where \(\phi\) and \(\psi\) are any two functions, then \[ x^2\frac{\partial^2 z}{\partial x^2} + 2xy\frac{\partial^2 z}{\partial x\partial y} + y^2\frac{\partial^2 z}{\partial y^2} = 0. \]

A rhombus of uniform rods \(ABCD\) freely jointed together rests symmetrically with \(AC\) horizontal, the rods \(AB, BC\) resting on two smooth pegs at \(E\) and \(F\). By means of the principle of virtual work or otherwise shew that in equilibrium the vertical through \(A\) and the horizontal through \(B\) meet on the normal at \(E\) to \(AB\). Find also the reactions at \(A\), \(B\) and \(E\).

A uniform circular cylinder rests on a rough horizontal plane (coefficient of friction \(\mu_1\)). A second cylinder rests partly on the former, touching it along one generator (the coefficient of friction between them being \(\mu_2\)), and partly on an inclined plane of inclination \(2\beta\), touching this along another generator (the coefficient of friction here being \(\mu_3\)). The plane through the axes of the cylinders makes with the vertical an angle \(2\alpha\). Shew that for equilibrium to be possible \[ \mu_3 > \tan\beta, \quad \mu_2 > \tan\alpha, \] and \[ \mu_1 \ge \frac{\tan\beta\tan\alpha}{\tan\beta+\frac{W}{W'}(\tan\alpha+\tan\beta)}, \] where \(W\) and \(W'\) are the weights of the upper and lower cylinders, respectively.

The corners \(A, B, C, D\) of a rigid rectangular platform are attached to and rest in a horizontal plane on four vertical springs of slight compressibility, the compressions of which for unit load are \(\lambda_1, \lambda_2, \lambda_3, \lambda_4\) respectively. Shew that a weight \(P\) placed on the platform at \(A\) will produce loads on the four springs given by \[ \frac{P_1}{\lambda_2+\lambda_3+\lambda_4} = \frac{P_2}{-\lambda_3} = \frac{P_3}{\lambda_4} = \frac{P_4}{-\lambda_2} = \frac{P}{\lambda_1+\lambda_2+\lambda_3+\lambda_4}. \] Note: The image for the above relations is blurry, this is a best-effort transcription. Hence shew that a weight \(W\) placed on the platform at a point \(O\), the distances of which from the parallel sides \(AB, DC\) are \(a_1, a_2\) and from the sides \(AD, BC\) are \(b_1, b_2\), will produce loads on the springs equal to \[ \frac{Wa_2b_2}{ab}-w, \quad \frac{Wa_2b_1}{ab}+w, \quad \frac{Wa_1b_1}{ab}-w, \quad \frac{Wa_1b_2}{ab}+w, \] where \[ w = \frac{W}{W_r} = \frac{\lambda_1\lambda_3 a_2 b_2 - \lambda_2\lambda_4 a_1 b_1 + \lambda_3\lambda_4 a_1 b_2 - \lambda_1\lambda_2 a_2 b_1}{(\lambda_1+\lambda_2+\lambda_3+\lambda_4)ab}, \] where \(a=a_1+a_2\) and \(b=b_1+b_2\).

You are given a number of unequal particles and a number of unequal pieces of elastic string. Explain how, from a knowledge of the accelerations produced in the particles by means of the strings and without any prior assumptions as to mass or force, to establish Newton's Law of Motion, \(P=Mf\), and Hooke's Law for an elastic string.

Shew that the time of swing of a simple pendulum is independent of the amplitude if the cube of the ratio of the amplitude to the length is neglected. A pendulum of length 32 feet is drawn aside a distance of 1 foot and the bob is then projected towards the position of equilibrium with a velocity of 1 foot per second. Find the point at which the bob will first come to rest and the time from the moment of projection to that point.