The Complete Equilibrium

Q. A uniform rod of length $l$ and mass $m$ is hung from two strings of equal length from a ceiling as shown in figure. Determine the tensions in the strings?

1. $T_A=\frac{2mg}{3};T_B=\frac{mg}{3}$
2. $T_A=\frac{mg}{3};T_B=\frac{2mg}{3}$
3. $T_A=T_B=\frac{mg}{2}$
4. $T_A=\frac{mg}{3},T_B=\frac{mg}{4}$

Q. A uniform metre scale balances horizontally on a knife edge placed at $55\text{cm}$ mark when a mass of $25\text{g}$ is suspended from one end, then mass of the scale is

1. $200\text{g}$
2. $225\text{g}$
3. $350\text{g}$
4. $275\text{g}$

Q. A system consists of three masses $m_1,m_2$ and $m_3$ connected by a string passing over a pulley $P$. The mass $m_1$ hangs freely and $m_2$ and $m_3$ are on a rough horizontal table (the coefficient of friction $=\mu$). The pulley is frictionless and of negligible mass. The downward acceleration of mass $m_1$ is (Assume $m_1=m_2=m_3=m$)

1. $\frac{g(1-2\mu )}{9}$
2. $\frac{2g\mu }{3}$
3. $\frac{g(1-2\mu )}{3}$
4. $\frac{g(1-2\mu )}{2}$

Q. A cubical block of mass $M$ and edge $a$ slides down a rough inclined plane of inclination $\theta$ with a uniform velocity. Choose the correct statement(s):

1. Torque of normal force on the block about centre is $\frac{Mga\sin \theta }{2}$
2. Torque of normal force on the block about the centre is $\frac{Mga\cos \theta }{2}$
3. Normal reaction force shifts by a distance $\frac{a\tan \theta }{2}$
4. Normal reaction force shifts by a distance $\frac{a\tan \theta }{4}$

Q. Two weights $w_1$ and $w_2$ are suspended from the ends of a light string passing over a smooth fixed pulley. If the pulley is pulled up at an acceleration $g$, the tension in the string will be

1. $\frac{2w_1{w}_2}{w_1+w_2}$
2. $\frac{w_1{w}_2}{w_1+w_2}$
3. $\frac{w_1{w}_2}{2(w_1+w_2)}$
4. $\frac{4w_1{w}_2}{w_1+w_2}$

Q. Two uniform rods of equal length but different masses are rigidly joined to form an $L$ - shaped body which is pivoted about point $\text{O}$ as shown. If in the shown configuration, the body is in equillibrium, the ratio $M/m$ will be.

1. $\sqrt{3}$
2. $\sqrt{2}$
3. $\frac{1}{\sqrt{3}}$
4. $2$

Q. A uniform rod of mass $2\text{kg}$ is hanging from a thread attached at the midpoint $\left(\text{O}\right)$ of the rod. A block of mass $m=6\text{kg}$ hangs at the left end of the rod and a block of mass $M$ hangs at the right end at a distance of $30\text{cm}$ from the mid point $\left(\text{O}\right)$. If the system is in equilibrium, calculate the mass $\left(M\right)$, given that overall length of the rod is $100\text{cm}$. Take $g=10{\text{m/s}}^2$.

1. $5\text{kg}$
2. $10\text{kg}$
3. $8\text{kg}$
4. $12\text{kg}$

Q. In figure, the bar is uniform and weighing $500\text{N}$. How large must $W\text{in N}$ be if $T_1$ and $T_2$ are to be equal?

Q. A square plate is hinged as shown in figure and it is held stationary by means of a light thread as shown in figure. Then find out the force exerted by the hinge.

1. $0$
2. $mg$
3. $mg-T$
4. $mg+T$

Q.

A uniform cylinder of radius R is spinned about its axis to the angular velocity ${\omega }_0$ and then placed into a corner, see the figure. The coefficient of friction between the corner walls and the cylinder is ${\mu }_k$. How many turns will the cylinder accomplish before it stops?

1. 
2. 

3. 

4. 

Q. A slab is subjected to two forces $\overrightarrow{F_1}\text{and}\overrightarrow{F_2}$ of same magnitude $F$ as shown in the figure. Force $\overrightarrow{F_2}$ is in $XY-\text{plane}$ while force $\overrightarrow{F_1}$ acts along $z-\text{axis}$ at the point $(2\hat{i}+3\hat{j})$. The moment of these forces about point $O$ will be:

1. $(3\hat{i}-2\hat{j}+3\hat{k})F$
2. $(3\hat{i}-2\hat{j}-3\hat{k})F$
3. $(3\hat{i}+2\hat{j}-3\hat{k})F$
4. $(3\hat{i}+2\hat{j}+3\hat{k})F$

Q. A uniform rod of mass $m$ and length $l$ hinged at point $\text{H}$ can rotate in a vertical plane about a smooth horizontal axis. Find the force exerted by the hinge just after the rod is released from rest, from an initial position making an angle of ${37}^{\circ }$ with horizontal.

1. $\sqrt{1}0mg$
2. $\frac{\sqrt{1}0}{5}mg$
3. $\frac{3mg}{5}$
4. $\frac{mg}{5}$

Q. A solid disc of radius of $a$ and mass $m$ rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the disc will be $\frac{2}{b}g\sin \theta$ where $b$ is(Round off to Nearest Integer) ($g=$ acceleration due to gravitiy; $\theta =$angle as shown in figure)

Q. A metal rod of length $50\text{cm}$ having mass $2\text{kg}$ is supported on two edges placed $10\text{cm}$ from each end. A $3\text{kg}$ load is suspended at $20\text{cm}$ from one end. Find the reactions at the edges (take $g=10{\text{m/s}}^2$)

1. $30\text{N},10\text{N}$
2. $20\text{N},20\text{N}$
3. $30\text{N},30\text{N}$
4. $30\text{N},20\text{N}$

Q.

A spring of spring constant $k$ is placed horizontally on a rough horizontal surface. It is compressed against a block of mass $m$ which is placed on a rough surface, so as to store maximum energy in the spring. If the coefficient of friction between the block and the surface is $\mu$, the potential energy stored in the spring is: (block does not slide due to force of spring.)

Q. Which of the following conditions are necessary for a rigid body to be in mechanical equilibrium?
Condition 1: Net external force acting on body is zero.
Condition 2: Net external torque acting on body is zero.
Condition 3: Internal forces and torque due to internal forces must be absent.

1. Both condition 1 and condition 2 are necessary.
2. Both condition 2 and condition 3 are necessary.
3. Both condition 1 and condition 3 are necessary.
4. All conditions are necessary.

Q. Two bodies of masses $2\text{kg}$ and $4\text{kg}$ are tied to the ends of a massless string. The string passes over a pulley which is frictionless as shown in figure. If the system is released from rest, then tension in the string connecting the bodies will be $(g=10{\text{m/s}}^2)$

1. $80\text{N}$
2. $\frac{80}{3}\text{N}$
3. $20\text{N}$
4. $\frac{20}{3}\text{N}$

Q. Two identical ladders are arranged as shown in figure. Mass of each ladder is $M$ and length is $L$. A load of mass $m$ has been attached at the touching point $\left(\text{O}\right)$ of the ladders, as shown in figure. The system is in equilibrium. Find the magnitude of static frictional force acting at either point $A$ and $B$.

1. $\left(\frac{M+m}{2}\right)g\tan \theta$
2. $\left(\frac{M+m}{2}\right)g\cot \theta$
3. $\left(\frac{M+m}{2}\right)g\cos \theta$
4. $\left(\frac{M+m}{2}\right)g\sin \theta$

Q.

A uniform rod of mass M and length L is placed in a horizontal plane with one end hinged about the vertical axis. A horizontal force $F=\frac{Mg}{2}$ is applied at a distance $\frac{5L}{6}$ from the hinged end. The angular acceleration of the rod will be

1. 

2. 

3. 

4. 

Q. A solid cylinder of mass $m$ is kept in balance on a fixed incline of angle $\alpha ={37}^{\circ }$ with the help of a thread fastened to its jacket. The cylinder does not slip. What will be the value of force $F$ that is required to keep the cylinder in balance when the thread is held vertically?

1. $\frac{mg}{8}$
2. $\frac{2mg}{7}$
3. $\frac{3mg}{5}$
4. $\frac{3mg}{8}$

Q. A ladder $AB$ is supported by a smooth vertical wall and rough horizontal floor as shown. A boy starts moving from $A$ to $B$ slowly. The ladder remains at rest, then pick up the correct statement(s):

1. Magnitude of normal reaction by wall on ladder at point $B$ will increase.
2. Magnitude of normal reaction by wall on ladder at point $B$ will decrease.
3. Magnitude of normal reaction by floor on ladder at point $A$ will remain unchanged.
4. Magnitude of frictional force by the floor on ladder at point $A$ will increase.

Q. In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle $\theta$ with the horizontal floor. The coefficient of friction between the wall and the ladder is $\mu$ and that between the floor and the ladder is ${\mu }_2.$ The normal reaction of the wall on the ladder is $N_1$ and that of the floor is $N_2.$ If the ladder is about to slip, then

1. ${\mu }_1=0{\mu }_2\ne 0and{N}_{2}tan\theta =mg/2$
2. ${\mu }_1\ne 0{\mu }_2=0and{N}_{1}tan\theta =mg/2$
3. ${\mu }_1\ne 0{\mu }_2\ne 0and{N}_2=\frac{mg}{1+{\mu }_1{\mu }_2}$
4. ${\mu }_1=0{\mu }_2\ne 0and{N}_{1}tan\theta =mg/2$

Q.

A uniform rod of length l and mass m is hung from two strings of equal length from a ceiling as shown in figure. The tensions in the strings are

1. 

2. 

3. 

4. 

Q. An object is said to be in complete equilibrium when

1. There is no net external force acting on the object
2. The total clockwise moment about any reference point is equal to the total anti-clockwise moment about the same point
3. The object moves with a constant angular velocity.
4. The object moves with a constant acceleration

Q.

A uniform metal chain is placed on a rough table such that one end of the chain hangs down over the edge of the table. When one-third of its length hangs over the edge, the chain starts sliding. Then, the coefficient of static friction is

1. $3/4$

2. $1/4$

3. $2/3$

4. $1/2$

Q. Assuming frictionless contact everywhere, determine the magnitude of external horizontal force $P$ applied at the lower end for equilibrium of the rod as shown in figure. The rod is uniform and its mass is ${}^{\prime }{m}^{\prime }$.

1. $\frac{mg}{2}\cot \theta$
2. $\frac{mg}{2}\sin \theta$
3. $mg\cos \theta$
4. $mg\tan \theta$

Q. A horizontal uniform rod of mass ${}^{\prime }{m}^{\prime }$ has its left end hinged to the fixed incline plane, while its right end rests on the top of a uniform cylinder of mass ${}^{\prime }{m}^{\prime }$ which in turn is at rest on the fixed inclined plane as shown. The coefficient of friction between the cylinder and rod, and between the cylinder and inclined plane, is sufficient to keep the cylinder at rest.


The magnitude of normal reaction exerted by the rod on the cylinder is

1. $\frac{mg}{4}$
2. $\frac{mg}{3}$
3. $\frac{mg}{2}$
4. $\frac{2mg}{3}$

Q. If the earth were a perfect sphere of radius $6.4\times {10}^6\text{m}$ rotating about its axis with the period of one day $(8.64\times {10}^4\text{sec})$, what is the difference in acceleration due to gravity on poles and the equator?

1. $338\times {10}^{-6}{\text{m/s}}^2$
2. $338\times {10}^{-4}{\text{m/s}}^2$
3. $338\times {10}^{-2}{\text{m/s}}^2$
4. None of these

Q. A rigid massless beam is balanced by a body of mass $4m$ on the left hand side and a pulley-mass system on the right hand side. The value of $\frac{x}{y}$ is :
(Consider the rope and pulley to be massless).

1. $\frac{7}{6}$
2. $\frac{2}{3}$
3. $1$
4. $\frac{11}{12}$

Q. A rod of mass $m$ is placed on two supports as shown. Calculate the normal reaction of each support.

1. $N_1=\frac{1}{3}mg,N_2=\frac{2}{3}mg$
2. $N_1=\frac{2}{3}mg,N_2=\frac{1}{3}mg$
3. $N_1=\frac{mg}{2},N_2=\frac{mg}{2}$
4. $N_1=\frac{mg}{4},N_2=\frac{3mg}{4}$