Assuming the principle of Virtual Work deduce the conditions of equilibrium of a system of coplanar forces acting on a rigid body.
State Newton's Laws of Motion. Describe an experimental verification of that part of the second Law of Motion expressed by the equation \(P=mf\).
Prove that the envelope of all the paths described by heavy particles projected from a given point with a given velocity is a paraboloid of revolution having the point as focus. Through this focus any plane is drawn. Shew that the section of the paraboloid by the plane is an ellipse or parabola.
Three light wires \(DA, AB, BC\) each of length \(2a\) are jointed together at \(A\) and \(B\) so that \(ABCD\) is a parallelogram and \(AD, BC\) are free to turn about the fixed points \(D, C\) in the plane of the figure. If the joints carry equal particles each of mass \(m\), find the angular velocity of \(AD\) just after impact if its middle point be struck by a blow \(I\) at right angles to it.
Define the hodograph and prove one of its properties. A particle describes a circle freely under the action of a constant force (not tending to the centre). Prove that the hodograph is the curve \(r^2=c^2\sin\psi\) where \(r\) is a radius vector from the origin to a point on the curve and \(\psi\) is the angle the tangent at the point makes with \(r\).
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.