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 $l(>4a)$ 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 ${\displaystyle \frac{W(l-3a)}{2\sqrt{l^2-6la+8a^2}}}$.

[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 ${\displaystyle \frac{1}{2}}W\sqrt{4+{\tan }^2\alpha }$, 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 $OA$, $OB$ 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 $\tan \theta ={\displaystyle \frac{P+Q}{P\cot \alpha -Q\cot \beta }}$.

[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 $\mathrm{A}\mathrm{B}$, $\mathrm{B}\mathrm{C}$, $\mathrm{C}\mathrm{D}$, $\mathrm{D}\mathrm{E}$, $\mathrm{E}\mathrm{F}$ and $\mathrm{F}\mathrm{A}$ are each of weight $\mathrm{W}$ and are freely jointed at their extremities so as to form a hexagon, the rod $\mathrm{A}\mathrm{B}$ is fixed in a horizontal position and the middle points of $\mathrm{A}\mathrm{B}$ 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 $\mathrm{k}\mathrm{g}$ 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 $\mathrm{k}\mathrm{g}$ begins to ascend the ladder. If the string can bear a tension of 10 $\mathrm{k}\mathrm{g}$ 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 $\cos \theta ={\displaystyle \frac{h}{2a}}\sqrt{{\displaystyle \frac{\sigma }{\rho -\sigma }}}$.

[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 $\mathrm{A}\mathrm{C}$ 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 $\overrightarrow{R}$ acting at the point $(4\hat{i}+2\hat{j})$ together with a couple $\overrightarrow{G}$ 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 $O{O}^{\prime }$, $A{A}^{\prime }$, $B{B}^{\prime }$ and $C{C}^{\prime }$ are its diagonals. Along $O{B}^{\prime }$, $O^{\prime }A$, $BC$ and $C^{\prime }{A}^{\prime }$ act forces equal to $P$, $2P$, $3P$ and $4P$ respectively. Reduce the system to a force at $O$ together with a couple.

[2001, 15M]