PYQs

1) A square \(ABCD\), the length of whose sides is \(a\), is fixed in a vertical plane with two of its sides horizontal. An endless string of length \(l(>4 a)\) passes over four pegs at the angles of the board and through a ring of weight \(W\) which is hanging vertically. Show that the tension of the string is \(\dfrac{W(l-3 a)}{2 \sqrt{l^{2}-6 l a+8 a^{2}}}\).

[2016, 20M]

2) A uniform rod \(AB\) of length 2 a movable about a hinge at \(A\) rests with other end against a smooth vertical wall. If \(\alpha\) is inclination of the rod to the vertical, prove that the magnitude of reaction of the hinge is \(\dfrac{1}{2} W \sqrt{4+\tan ^{2} \alpha}\), where \(W\) is the weight of the rod.

[2016, 15M]

3) Two weights \(P\) and \(Q\) are suspended from a fixed point \(O\) by strings \(O A\), \(O B\) and are kept apart by a light rod \(AB\). If the strings \(OA\) and \(OB\) make angles \(\alpha\) and \(\beta\) with the rod \(AB\), show that the angle \(\theta\) which the rod makes with the vertical is given by \(\tan \theta=\dfrac{P+Q}{P \cot \alpha-Q \cot \beta}\).

[2016, 15M]

4) A rod of 8 kg movable in a vertical plane about a hinge at one end, another end is fastened a weight equal to half of the rod, this end is fastened by a string of length \(l\) to a point at a height \(b\) above the hinge vertically. Obtain the tension in the string.

[2015, 10M]

5) Six equal rods \(\mathrm{AB}\), \(\mathrm{BC}\), \(\mathrm{CD}\), \(\mathrm{DE}\), \(\mathrm{EF}\) and \(\mathrm{FA}\) are each of weight \(\mathrm{W}\) and are freely jointed at their extremities so as to form a hexagon, the rod \(\mathrm{AB}\) is fixed in a horizontal position and the middle points of \(\mathrm{AB}\) and \(DE\) are joined by a string. Find the tension in the string.

[2013, 15M]

6) A ladder of weight \(W\) rests with one end against a smooth vertical wall and the other end rests on a smooth floor. If the inclination of the ladder to the horizon is \(60^{\circ}\), find the horizontal force that must be applied to the lower end to prevent the ladder from slipping down.

[2011, 10M]

7) A uniform rod \(AB\) is movable about a hinge at \(A\) and rests with one end in contact with a smooth vertical wall. If the rod is inclined at an angle of \(30^{\circ}\) with the horizontal, find the reaction at the hinge in magnitude and direction.

[2009, 12M]

8) A ladder of weight 10 \(\mathrm{kg}\) rests on a smooth horizontal ground leaning against a smooth vertical wall at an inclination \(\tan ^{-1} 2\) with the horizon and is prevented from slipping by a string attached at its lower end, and to the junction of the floor and the wall. A boy of weight 30 \(\mathrm{kg}\) begins to ascend the ladder. If the string can bear a tension of 10 \(\mathrm{kg}\) wt, how far along the ladder can the boy rise with safety?

[2008, 15M]

9) A uniform rod of length \(2a\) can turn freely about one end, which is fixed at a height \(\mathrm{h}(<2 \mathrm{a})\) above the surface of the liquid. If the densities of the rod and liquid be \(\rho\) and \(\sigma\), show that the rod can rest either in a vertical position or inclined at an angle \(\theta\) to the vertical such that \(\cos \theta=\dfrac{h}{2 a} \sqrt{\dfrac{\sigma}{\rho-\sigma}}\).

[2006, 15M]

10) If a number of concurrent forces be represented in magnitude and direction by the sides of a closed polygon, taken in order, then show that these forces are in equilibrium.

[2005, 12M]

11) A uniform bar \(AB\) weighs 12 \(\mathrm{N}\) and rests with the part \(\mathrm{AC}\) of length 8 \(\mathrm{m}\) on a horizontal table and the remaining part \(CB\) projecting over the edge of the table. If the bar is on the point of overbalancing when a weight of 5 \(\mathrm{N}\) is placed on it at a point 2 \(\mathrm{m}\) from \(\mathrm{A}\) and a weight of 7 \(\mathrm{N}\) is hung from \(\mathrm{B}\), find the length of \(AB\).

[2004, 15M]

1) On a rigid body, the forces \(10(\hat{i} +2\hat{j}+2\hat{k})N\), \(5(-2\hat{i} -\hat{j}+2\hat{k})N\) and \(6(2\hat{i} +2\hat{j}-\hat{k})N\) are acting at points with position vectors \(\hat{i}-\hat{j}\), \(2\hat{i}+5\hat{k}\) and \(4\hat{i}-\hat{k}\) respectively. Reduce this system to a single force \(\vec{R}\) acting at the point \((4\hat{i}+2\hat{j})\) together with a couple \(\vec{G}\) whose axis passes through this point. Does the point \((4\hat{i}+2\hat{j})\) lie on the central axis?

[2009, 15M]

2) \(OA\), \(OB\) and \(OC\) are edges of a cube of side \(a\), and \(OO'\), \(AA'\), \(BB'\) and \(CC'\) are its diagonals. Along \(OB'\), \(O'A\), \(BC\) and \(C'A'\) act forces equal to \(P\), \(2P\), \(3P\) and \(4P\) respectively. Reduce the system to a force at \(O\) together with a couple.

[2001, 15M]