Problems

Prove that the general equation of a circle in areal coordinates is \[ (x+y+z)(t_1^2x+t_2^2y+t_3^2z) - (a^2yz+b^2zx+c^2xy) = 0, \] where \(a, b, c\) are the lengths of the sides of the triangle of reference \(ABC\) and \(t_1, t_2, t_3\) are the lengths of the tangents from \(A, B, C\) to the circle.

\(n\) equal, uniform, straight, smoothly jointed rods \(A_0A_1, A_1A_2, A_2A_3, \dots, A_{n-1}A_n\), are suspended from the end \(A_0\) while the end \(A_n\) can move on a fixed smooth vertical wire. The whole system is in equilibrium in a vertical plane with the rods inclined to the horizontal at angles \(\theta_1, \theta_2, \theta_3, \dots, \theta_n\) respectively. Prove that \[ (2n-1)\cot\theta_1 = (2n-3)\cot\theta_2 = (2n-5)\cot\theta_3 = \dots = \cot\theta_n = \frac{2R}{W}, \] where \(W\) is the weight of each rod and \(R\) the reaction at \(A_n\).

Define a couple and establish the principal properties of a couple. The figure represents the horizontal section through a door \(OA\) which can move between the two extreme perpendicular positions \(OX, OY\) about hinges in the vertical straight line through \(O\). \(AY\) is a light extensible string which obeys Hooke's Law. A spring exerts a restoring couple proportional to \(\theta\) and the hinges produce a constant frictional couple \(F\) when the door is in motion. It is found that the work done in slowly opening the door from \(OX\) to \(OY\) is equal to the work done in slowly closing the door again, whereas the work done in slowly half opening the door is \(\frac{1}{n}\)th of the work done in slowly closing the door from this position. Shew that if the string \(AY\) were cut, then the door would remain ajar provided \(\theta\) were not greater than \(\frac{3}{8}\frac{n+1}{n-1}\pi\). It may be assumed that \(n > 7\) and that the unstretched length of the string lies between \(2OA\sin\frac{\pi}{8}\) and \(2OA\sin\frac{\pi}{4}\).

Prove that a uniform solid elliptic cylinder can be in equilibrium on a rough inclined plane with its generators horizontal provided the eccentricity \(e\) of the normal cross-section of the cylinder is not less than \(\tan\alpha (2\operatorname{cosec}\alpha - 2)^{1/2}\), where \(\alpha\) is the inclination of the plane to the horizontal. If this condition is satisfied how many distinct positions of equilibrium exist? Determine the stability, or otherwise, of any possible positions of equilibrium in the case \[ \tan\alpha = \frac{\sqrt{2}}{8}, \quad e = \frac{\sqrt{2}}{2}. \]

Define the bending moment and shearing stress at a point of a beam. Draw the bending moment and shearing stress diagrams for a uniform horizontal beam \(AC\) of weight \(W\) which is freely hinged at the end \(A\) and rests on a support at \(B\) where \(AB=2BC\), and which supports a weight \(W\) at \(C\).

State Newton's Laws of Motion and shew how they give rise to the equation \(P=mf\) and to the absolute and gravitational units of force. A particle of variable mass is in motion in a straight line under the action of a force whose magnitude in absolute units at any instant is twice the product of the mass of the particle and its acceleration at that instant. Prove that the impulse of the force in any interval is proportional to the change in the square of the velocity in that interval. It may be assumed that the particle does not come to rest at any instant in the interval considered.

A lift moves vertically upwards from rest with uniform acceleration \(f( < g)\) and as it starts to move a ball is dropped on to the floor of the lift. If the lift overtakes the ball when the latter is at the highest point (in space) of its first bounce, shew that the coefficient of restitution between the ball and the floor of the lift is \(\frac{f}{g-f}\) and that the time that elapses between the instant of starting and the instant of overtaking is \(\frac{\sqrt{2h(g+f)}}{g-f}\) where \(h\) is the initial height of the ball above the floor of the lift. (Note: The scanned document's formula for the coefficient of restitution is unclear, this transcription uses the physically derived correct value.)

The time taken by a shell of mass \(m\) fired with speed \(V\) at an angle \(\alpha\) to the horizontal to reach the highest point of its trajectory is \(t\) seconds. \(\frac{3t}{2}\) seconds after firing, the shell is split into two parts of equal mass by an explosion which increases the energy of the subsequent motion by \(\frac{mV^2}{8}\). Immediately after the explosion it is observed that the horizontal velocity of one part has been increased and its vertical velocity annulled and that both parts continue to move in the same vertical plane as that in which the motion was taking place before the explosion. Prove that the two parts strike the horizontal plane through the point of projection at a distance apart \[ \frac{V^2\sin 2\alpha}{8g}(3\sqrt{3}+2-\sqrt{7}). \]

Prove that, if \(p/q\) is a fraction in its lowest terms, then integers \(r\) and \(s\) can be found such that \(qr-ps=1\). Prove that, if \(p/q\) and \(r/s\) are fractions such that \(qr-ps=1\), then the denominator of any fraction whose value lies between \(p/q\) and \(r/s\) is at least \(q+s\).

Prove that \[ a^2+b^2+c^2-bc-ca-ab = (a+\omega b+\omega^2 c)(a+\omega^2 b + \omega c), \] where \(\omega\) is a complex cube root of 1. Prove that, if \[ (b-c)^n+(c-a)^n+(a-b)^n \] is divisible by \(\Sigma a^2 - \Sigma bc\), then \(n\) is an integer not a multiple of 3. Prove that, if the same expression is divisible by \((\Sigma a^2 - \Sigma bc)^2\), then \(n\) is greater by one than a multiple of 3.