Two cylinders of unequal radii are placed with their axes parallel on a horizontal plane and a plank is laid across them with its length at right angles to the axes of the cylinders and its centre of gravity half-way between the points of contact. One cylinder is held fixed and the plane is sufficiently rough to prevent the other cylinder from slipping. Prove that if \(\mu\) is the coefficient of friction between the plank and either cylinder and if \(\alpha\) is the inclination of the plank to the horizontal, the system is in equilibrium if \(\mu > \frac{1}{2}(2\tan\alpha - \tan\alpha/2)\), and that the first cylinder can be held fixed by a couple of magnitude \(Wa\sin\alpha\), where \(W\) is the weight of the plank and \(a\) is the radius of that cylinder.
If \(A\) and \(B\) are points on a rod which is moving in any way in a plane, and if \(Oa\) and \(Ob\) represent the velocities of \(A\) and \(B\) at any instant, prove that \(ab\) is perpendicular to \(AB\). If \(C\) is any other point on the rod and if \(c\) divides \(ab\) in the same ratio as that in which \(C\) divides \(AB\), prove that \(Oc\) represents the velocity of \(C\) at the same instant. \par \(PQ, QR, RS\) are three rods in a plane jointed together at \(Q\) and \(R\), and with the ends \(P\) and \(S\) jointed to fixed supports. If a triangle \(Oqr\) is drawn with \(Oq, qr, ro\) perpendicular to \(PQ, QR, RS\) respectively for any position of the rods, prove that as the rods move through this position \(Oq\) and \(Or\) represent on the same scale the velocities of \(Q\) and \(R\).
A train starting from rest is uniformly accelerated until its velocity is 30 feet per second and then uniformly accelerated at half the previous rate until its velocity is 60 feet per second and after that the velocity remains uniform. The train takes 118 seconds to travel the first mile. Find the initial acceleration. \par If the train weighs 200 tons and the resistances to motion are equivalent to a back pull of 16 lbs. wt. per ton find the average horse-power and also the maximum horse-power at which the engine was working during the time it took to travel the first mile.
Prove that the path of a projectile under no forces but gravity is a parabola. \par An aeroplane is flying with constant velocity \(v\) and at constant height \(h\). Show that, if a gun is fired point blank at the aeroplane after it has passed directly over the gun and when its angle of elevation as seen from the gun is \(\alpha\), the shell will hit the aeroplane provided \(2(V\cos\alpha-v)v\tan^2\alpha = gh\), where \(V\) is the initial velocity of the shot, the path being assumed to be parabolic.
A particle of mass \(m\) moving with velocity \(u\) impinges on a particle of mass \(M\). If after the impact the component velocities of the mass \(m\) are \(u'\) and \(v'\) in directions along and perpendicular to the original direction of motion, find the component velocities of the mass \(M\) in these directions. \par If there is no loss of energy in the impact prove that the greatest value that \(v'\) can have is \(\frac{M}{M+m}u\), and that this occurs when the line of the impulse makes an angle of \(\frac{\pi}{4}\) with the original direction of motion.
A particle when hanging in equilibrium at the end of a light elastic string stretches it a distance \(a\). Prove that the period of vibration of the particle in a vertical line through its equilibrium position is the same as that of a simple pendulum of length \(a\). \par A light endless elastic string of unstretched length \(2b\) passes over two small smooth pegs on the same level distant \(b\) apart. A particle is attached to a point on the string and when the particle is in equilibrium the string forms the three sides of an equilateral triangle. Prove that the period of vibration of the particle in a vertical line is the same as that of a pendulum of length \(\frac{2\sqrt{3}}{7}b\).
The inscribed circle of the triangle \(ABC\) touches \(BC\) at \(D\), \(CA\) at \(E\) and \(AB\) at \(F\); \(P\) is the other extremity of the diameter of the circle through \(D\); \(BC\) is cut by \(AP\) at \(M\), by \(PE\) at \(Q\) and by \(PF\) at \(R\). Prove that \(M\) is the middle point of \(QR\).
Having given that \begin{align*} x+y+z &= a, \\ x^2+y^2+z^2 &= b^2, \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} &= \frac{1}{c}, \end{align*} determine \(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\) in terms of \(a, b, c\). \par Solve the equations \[ \left\{ \begin{array}{l} xyz = 1, \\ (x+1)(y+1)(z+1)=9, \\ (y+z)(z+x)(x+y) = 11\frac{1}{4}. \end{array} \right. \]
Find how many different numbers between 1000 and 10,000 can be formed with the digits 0, 1, 2, 3, 4, 5. Shew that the sum of the numbers so formed is 979,920.
The roots of the equation \[ x^3 - ax^2 + bx - c = 0, \] are the lengths of the sides of a triangle. \par Shew that the area of the triangle is \(\frac{1}{4}\{a(4ab-a^3-8c)\}^{\frac{1}{2}}\), and that if the triangle is right-angled then \[ a(4ab-a^3-8c)(a^2-2b)-8c^2=0. \]