Problems

\(AB, BC, CD, DE, EA\) are five equal uniform rods each of weight \(w\) and smoothly jointed. \(A\) is connected to \(C\) and \(D\) by two equal light inextensible strings, and the whole is suspended freely from \(A\). If the inclinations to the vertical of \(AB, AC\) are \(\theta, \phi\) respectively, shew that \(\sin(2\phi-\theta) + \sin\theta = \frac{1}{4}\). By use of the principle of virtual work, or otherwise, prove that the tension in each string is \[ w\{\sin\phi \text{cosec}(2\theta-2\phi) + \tfrac{1}{2}\sec\phi\}. \]

A particle moves in a straight line in such a manner that its velocity, \(t\) seconds after it is projected with velocity \(u\) from the point from which the distance \(s\) is measured, is \(ue^{at+bs}\), where \(a\) and \(b\) are positive constants. Prove that the time taken to attain velocity \(2u\) is \[ \frac{2}{a} \coth^{-1}\left(\frac{a}{bu}+3\right), \] and determine the distance travelled during this interval.

A smooth thin wire is bent into the shape of a semicircle of radius \(a\) and fixed in a vertical plane. A bead slides on the wire starting from rest at the higher end, and, after leaving the wire at its lower end, moves horizontally when at the level of the centre of the circle of which the semicircular wire is one half. Find, in terms of \(a\), the horizontal distance between the initial position of the bead and the point at which the bead when moving freely is at the same level as the lowest point of the wire.

Two equal particles are connected by a light inelastic string of length \(2l\). The particles are at rest on a smooth horizontal table at points \(A, B\) at a distance \(l\sec\phi\) apart when the particle at \(B\) is caused to move on the table with velocity \(V\) in a direction making an acute angle \(2\phi\) with the direction of \(AB\) produced. Shew that the particle which was at \(B\) initially is again moving parallel to its initial direction of motion after time \[ \frac{l}{V}\{\sec\phi + (\pi+2\phi)\text{cosec}\phi\}. \]

A light elastic string of unstretched length \(l\) passes through two smooth rings fixed at a distance \(l\) apart at the same horizontal level. The ends of the string are attached to a particle of mass \(m\) and when the system hangs in equilibrium the three portions of the string form the sides of an equilateral triangle. When in this position the particle receives a vertical impulse \(I\). Prove that the maximum rate at which work is done in the subsequent motion is \(\displaystyle \frac{I^2}{m} \left(\frac{g}{2l\sqrt{3}}\right)\).

Define the coefficient of restitution of two bodies. A smooth, thin, straight tube \(AB\) of length \(l\) is fixed in a position making an angle \(\beta\) with the vertical, the lower end \(B\) of the tube being closed. A particle slides from rest at \(A\) inside the tube and at the instant when it rebounds from the end \(B\), a second and equal particle begins to slide from rest at \(A\) inside the tube. Find the time that elapses before the particles collide, and shew that the velocity of the second particle immediately after colliding with the first is \[ \sqrt{gl\left\{\frac{1-4e^3-5e^4-e^5}{2e(2+e)}\right\}} \] down the tube, where \(e\) is the coefficient of restitution between the two particles and also between the first particle and the end of the tube, and \(2 > 1+2e > \sqrt{3}\).

A uniform straight rod of mass \(M\) and length \(l\) can turn about one end on a rough horizontal table. A spring exerts a restoring couple \(\mu^2 Mgl\theta\) when the rod makes an angle \(\theta\) with a standard position in the plane. \(\mu\) is the coefficient of friction between the rod and the table. If the rod is started from the standard position with angular velocity \(\omega\), find the least value of \(\omega\) for which the rod (a) eventually begins to return towards the standard position, (b) eventually passes through the standard position again.

Find necessary conditions to be satisfied by the coefficients \(a, b, c\) in order that \(ax^2 + 2bx + c\) may be positive for all real values of \(x\). Prove that these conditions are sufficient. Assuming that these conditions are satisfied, find, in terms of \(a, b, c, k\), the greatest value that \(h\) can have if \[ ax^2 + 2bx + c \ge h(x-k)^2 \] for all real values of \(x\).

Determine all sets of solutions \((x, y, z)\) of the equations \begin{align*} x + y + z &= a+b+c, \\ a^2x + b^2y + c^2z &= a^3+b^3+c^3, \\ a^3x + b^3y + c^3z &= a^4+b^4+c^4, \end{align*} where \(a, b, c\) are unequal, distinguishing the cases in which \(bc+ca+ab\) is different from or equal to zero.

Resolve \(x^{2n}+1\) into real quadratic factors, where \(n\) is a positive integer. Express \[ \frac{1}{x^{2n}+1} \] in partial fractions with these factors as denominators.