One end of a uniform rod of length \(l\) and weight \(w\) is freely jointed to a point in a smooth vertical wall, the other end of the rod is freely jointed to a point in the surface of a uniform sphere of weight \(W\) and radius \(a\) which rests against the wall. Prove that when there is equilibrium the inclinations \(\beta\) and \(\alpha\) to the vertical of the rod and of the radius of the sphere through the point of attachment of the rod are given by the equations \[ a\sin\alpha+l\sin\beta = a, \quad \left(\frac{w}{2}+W\right)\tan\beta = W\tan\alpha. \]
One end of a string is attached to a fixed point \(O\) and particles of masses \(m, m'\) are attached to the string at intervals at the points \(A, A'\) respectively. The system is held at rest with the points \(O, A, A'\) in a vertical plane and the strings \(OA, AA'\) taut. It is now released from rest. Prove that the tension in the string \(OA\) is initially \[ \frac{m(m+m')g\cos\alpha}{m+m'\sin^2\beta} \] where \(\alpha\) is the angle \(OA\) makes with the vertical and \(\beta\) is the angle between \(OA\) and \(AA'\).
Two equal light rods of length \(l\) are jointed freely to each other and have particles of equal weight attached to their free ends. The rods are placed symmetrically across a smooth circular cylinder of radius \(a\) with its axis horizontal so as to touch the cylinder and with the joint vertically above the axis. The system is released from rest when \(2\alpha\) is the angle between the rods. Prove that if \(2\beta\) is the angle between the rods when the system next comes to rest \[ \sin\alpha\sin\beta\tan\frac{1}{2}(\alpha+\beta) = a/l. \]
Two forces act along given straight lines \(OA, OB\) and are represented in magnitude by \(lOA, mOB\) respectively. Prove that their resultant acts along \(OC\) and is represented by \((l+m)OC\), where \(C\) is the point in \(AB\) such that \(lAC=mCB\). If \(ABCD, A'B'C'D'\) are any parallelograms, prove that the forces acting at a point represented in magnitude and direction by \(AA', B'B, CC', D'D\) are in equilibrium.
Investigate necessary and sufficient conditions for the equilibrium of a body acted on by three forces. A uniform rod \(AB\), length \(2a\), rests with one end \(B\) on a rough horizontal plane, for which the angle of friction is \(\lambda\). The other end \(A\) is fastened to a fine string which passes over a small pulley vertically above and at a distance \(h(>2a)\) from \(B\). Prove that before the rod begins to slip at \(B\) it may be lowered into a position in which its inclination to the vertical is \(\lambda+\sin^{-1}\left(\frac{h}{2a}\sin\lambda\right)\).
State the Principle of Virtual Work and prove it in the case of forces acting on a body in one plane. A rhombus \(ABCD\) is formed of four light rods smoothly jointed at their ends. \(A\) is connected with \(C\) and \(B\) with the middle point of \(CD\) by tight strings. Shew that if these strings be of equal length the tension of one of them is double that of the other.
Define angular velocity. A circle, centre \(C\), rolls with uniform angular velocity \(\omega\) on the outside of another equal fixed circle whose centre is \(O\). Prove that the angular velocity of \(OP\), where \(P\) is any point on the circumference of the rolling circle, is \(\frac{6\sin^2\theta}{1+8\sin^2\theta}\omega\), where \(2\theta\) is the angle \(OCP\).
State the principle of the conservation of linear momentum. Three particles, each of mass \(m\), are attached to a light string at the points \(A, B, C\). They are placed on a smooth horizontal plane so that the angle \(ABC\) is obtuse and the portions \(AB\) and \(BC\) of the string are taut. A blow \(P\) is given to the particle \(B\) in a direction perpendicular to \(AC\). Prove that the particle \(A\) begins to move with a velocity \[ \frac{P}{m}\frac{2\sin A-\cos B\sin C}{3+\sin^2B}. \]
A particle is projected from a given point with a given velocity in a vertical plane under gravity. Prove that the envelope of all the possible paths is a parabola, and find its focus and latus rectum. A particle is projected under gravity from the highest point of a sphere; find the least velocity of projection that it may just clear the sphere.
Define a simple harmonic motion. Find the period of such a motion and shew that it is independent of the amplitude. A particle of weight \(w\) is attached to one end of a light elastic string of modulus \(\lambda\) and natural length \(l\), the other end of the string being attached to a fixed point. The weight is let go from rest from the fixed point. Describe the subsequent motion and find the time until the particle first returns to the fixed point.