A uniform regular hexagonal lamina \(ABCDEF\) rests in a vertical plane with the sides \(AB\) and \(CD\) in contact with two fixed parallel smooth horizontal rods in the same horizontal plane. Shew that the only position of equilibrium is the one in which \(BC\) is horizontal, and that it is stable.
A candidate is examined in three papers to which are assigned \(n\), \(n\), and \(2n\) marks respectively. His total marks are \(3n+1\). Shew that there are \(\frac{1}{2}n(n+1)\) ways in which this may have happened.
Explain the application of graphical methods to determine the velocity, space described and energy acquired by a particle moving with known acceleration. The acceleration of a particle remains constant during consecutive intervals of time \(\tau\), but increases in arithmetical progression at the end of each interval. Shew that the space described in any time \(t\), which is an odd multiple of \(\tau\), is \(\frac{u+4w+v}{6}t\), where \(u\) and \(v\) are the initial and final velocities, \(w\) is the velocity at the middle of the time.
Air is to be compressed into a chamber of volume \(V\) by means of a pump. The pump has a cylinder of volume \(v\); it takes in air from the atmosphere, compresses it adiabatically, and forces it into the chamber through a non-return valve. Show that, neglecting the clearance of the pump, the pressure in the chamber after \(n\) strokes of the pump is \(\left(1 + n\frac{v}{V}\right)^\gamma\) times the pressure of the atmosphere. (\(\gamma\) is the ratio of specific heat at constant pressure to specific heat at constant volume.)
Prove that, if \(r < a-b\), there are eight normals to the ellipse \(x^2/a^2 + y^2/b^2=1\) which are tangents to the circle \(x^2+y^2=r^2\); and shew that the corresponding tangents to the ellipse touch one or other of the circles \(x^2+y^2=R^2\), where \[2R = \{(a+b)^2 - r^2\}^{\frac{1}{2}} \pm \{(a-b)^2-r^2\}^{\frac{1}{2}}.\]
A particle is projected from \(O\) in the direction \(OT\) with velocity \(V\), and at the same instant an equal particle is let fall from \(T\). Shew that they collide, and that if they coalesce after collision, the path described is that of a particle projected at the same instant from the middle point of \(OT\) in the same direction \(OT\) with velocity \(\frac{1}{2}V\).
Prove that the sum of the squares of the medians of a triangle \(ABC\) is \(\frac{3}{4}(a^2+b^2+c^2)\). If \(B=55^\circ\), \(C=23^\circ 30'\) and the median \(AD = 400\) feet, find to the nearest foot the lengths of the median \(BE\).
Prove that kinetic energy is always destroyed in the impact of inelastic particles. A mass \(M\) is attached to one end of a fine inextensible string, and particles each of mass \(m\) are tied to the string at regular intervals \(a\). The string is placed over a fixed smooth pulley and the particles coiled together on a table. Shew that if the mass falls freely through a distance \(a(M+m)(4M-m)/6Mm\) before the string becomes taut, the \((r+1)\)th particle is dragged off the table with velocity \[ \left\{ga \frac{M+mr}{M+m(r+1)} \frac{4M - m(2r+1)}{m}\right\}^{\frac{1}{2}}. \]
The following table gives the volume (\(v\)) of one pound of dry saturated steam at different pressures (\(p\)):
Prove that the two circles \(3x^2+3y^2+6ax = a^2\) and the hyperbola \(6x^2 - 3y^2 = 2a^2\) are so related that any pair are polar reciprocals with regard to the third.