Prove that the motion of a particle suspended from a fixed point by an inelastic string oscillating through a small angle is approximately simple harmonic. A simple pendulum 4 feet long swings through an arc of 3 inches. Find the time of a complete oscillation and the velocity at a distance of 1 inch from the centre of the arc.
A system of coplanar forces will reduce in general to a single force or a couple. Prove this and mention the exceptional case. Forces \(P, Q, R\) act along the sides \(BC, CA, AB\) of a triangle. Find the conditions that their resultant should be parallel to \(BC\) and determine its magnitude.
A rigid plane framework of five jointed bars forming two equilateral triangles \(BAC, CDA\) is in equilibrium under the action of three forces on the joints at \(A, B, D\). Prove that the directions of these forces must be parallel or concurrent lying in the plane of the framework. If the force at \(B\) be 4 cwts. acting perpendicularly to \(CD\) and the force at \(A\) be in the direction \(BA\), find the force at \(D\) and the stresses in the bars by a graphical method.
Define the moment of a force about (1) a point, (2) a straight line. A fixed smooth axis, inclined to the horizon at an angle \(\alpha\), runs through a heavy body so that it is free to slide along and to turn round the axis. If the body be kept in equilibrium by a vertical force acting on it, find the conditions of equilibrium.
Investigate the position of the centre of gravity of a homogeneous solid hemisphere. Find the centre of gravity of a semi-circular plate of radius \(a\) and of uniform small thickness such that the density at any point varies as \(\sqrt{a^2-r^2}\) where \(r\) is the distance of the point from the centre.
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 \(r\).