Shew that the period of revolution of a conical pendulum is \(2\pi\sqrt{\dfrac{h}{g}}\), where \(h\) is the height of the point of support above the circular path of the bob. Derive from this result the time of a small oscillation of a simple circular pendulum.
Define Kinetic Energy. State and prove the principle of energy for a particle moving in a straight line under the action of a constant force in that line.
Through \(O\), the intersection of the diagonals \(AC\) and \(BD\) of a quadrilateral \(ABCD\), a straight line is drawn parallel to \(AB\) to cut \(CD\) in \(E\) and the third diagonal in \(F\). Prove that \(EF=OE\).
Through a point \(K\) in the major axis of an ellipse a chord \(PQ\) is drawn; prove that the tangents at \(P\) and \(Q\) intersect the line through \(K\) at right angles to the major axis in points equidistant from \(K\).
Prove that all spheres which cut orthogonally a system of spheres having a common plane of intersection pass through two fixed points.
Prove the convergency of the series whose \(n\)th term is \(\dfrac{1 \cdot 3 \cdot 5 \dots (2n-1)}{3^{n+1} \cdot n+2}\) and find the sum of an infinite number of terms of the series.
Prove that \(\Sigma[\cos 2A-\cos(B+C)](\cos B-\cos C) = \Sigma\sin(C-B)(\sin B+\sin C)\).
Two straight rulers with inches marked on them are laid across one another at a given angle so that the zero points do not coincide. Shew that perpendiculars drawn to the rulers at points having the same marks intersect on a line parallel to the bisector of the angle between the rulers.
Through a point \(O\) any two lines are drawn to cut, in \(P, Q\) and \(P', Q'\), any conic which touches two fixed lines through \(O\) at given points. Prove that \(PP'\) meets \(QQ'\) on a fixed line.
A pentagon \(ABCDE\) is formed of rods whose weight is \(w\) per unit length. The rods are freely jointed together and stand in a vertical plane with the lowest rod \(AB\) fixed horizontally, while the joints \(C, E\) are connected by a string. If \(BC\) and \(AE\) are of length \(a\), \(CD\) and \(DE\) of length \(b\) and the angles at \(A\) and \(B\) are each \(120^\circ\) and the angles at \(C\) and \(E\) are each \(90^\circ\), shew that the tension of the string is \(w\dfrac{a+5b}{2\sqrt{3}}\).