State the laws of statical friction. A heavy circular hoop is hung over a rough peg. A weight equal to that of the hoop is attached to it at a given point. Find the coefficient of friction between the peg and the hoop so that the system may hang in equilibrium whatever point of the hoop is placed in contact with the peg.
State the principle of Virtual Work, and prove it for a system of coplanar forces acting on a rigid body. A parallelogram \(ABCD\) is formed of uniform heavy rods freely jointed at their extremities. \(AB\) is held fixed in a horizontal position and the parallelogram is maintained in its form so that \(ADC\) is an acute angle \(\alpha\) by means of a string joining \(A\) to a point \(P\) in \(DC\). Prove that the tension of the string is \(W \cdot AP\cot\alpha/DP\), where \(W\) is half the weight of the parallelogram.
A particle is projected with velocity \(V\), from a point on an inclined plane, at an angle \(\beta\) to the line of greatest slope and in the vertical plane containing this line. Find the range on the inclined plane if its inclination to the horizontal is \(\alpha\). If \(\alpha\) is \(45^\circ\) and \(\beta\) is \(15^\circ\), shew that the direction of motion of the particle when it strikes the inclined plane makes an angle of \(30^\circ\) with the line of greatest slope.
State Newton's Laws of Motion and deduce the equation \(P=mf\). A particle of mass \(m\) slides down the face of a smooth inclined plane of mass \(M\) and inclination \(\alpha\) which is free to slide on a smooth horizontal plane. Prove that the acceleration of \(M\) is \(mg\sin\alpha\cos\alpha/(M+m\sin^2\alpha)\). Also find the pressure between the particle and the inclined plane.
Prove that \(v^2/r\) is the acceleration towards the centre of a particle moving in a circle with velocity \(v\). A heavy particle is placed inside a smooth circular tube fixed in a vertical plane. The particle is slightly displaced from rest at the highest point of the tube, prove that in the subsequent motion the pressures between it and the tube as it passes the extremity of the horizontal diameter and the lowest point of the tube are as 2:5.
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