The Complete Equilibrium

Q.

A carnival merry-go-round rotates about a vertical axis at a constant rate. A man standing on the edge has a constant speed of 3 m/s and a centripetal acceleration $\overrightarrow{a}$ of magnitude 2 m/s². Position vector $\overrightarrow{r}$ locates him relative to the rotation axis.

What is the magnitude of $\overrightarrow{r}$ ? What is the direction of $\overrightarrow{r}$ when $\overrightarrow{a}$ is directed due east?

1. 1.5 m, east

2. 4.5 m, east

3. 1.5 m, west

4. 4.5 m, west

Q. A non-uniform bar of weight W is suspended at rest by two strings of negligible weight as shown in Fig.7.39. The angles made by the strings with the vertical are 36.9° and 53.1° respectively. The bar is 2 m long. Calculate the distance d of the centre of gravity of the bar from its left end.

Q.

One fourth length of a uniform rod of length 2l and mass m is placed on a horizontal table and the rod is held horizontal. The rod is released from rest. Find the normal reaction on the rod as soon as the rod is released

1. 

2. 

3. 

4. 

Q. A block of mass $M=50\text{kg}$, base $b=1\text{m}$ and height $h=3\text{m}$ is kept on a rough inclined surface with coefficient of static friction $\mu =0.8$ as shown in figure. The angle of inclination with horizontal is ${37}^{\circ }$. Determine whether the block slides down or topples over.

1. slides down
2. topples over
3. remains in complete equilibrium
4. cannot be determined.

Q. A uniform shell explodes into three equal parts.Two of the parts fly off perpendicular to each other with velocities 9m/s and 12m/s.Find the velocity of the third part.

Q. 55.A disc is in the condition of pure rolling Honda horizontal surface velocity of centre of mass is Vo. If the radius of the disc is R, then the speed of point P is( OP =R/3 )

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

Q. Calculate the force required to lift a load of $60\text{N}$, placed at a distance of $3\text{m}$ from the fulcrum, if the effort is applied at a distance of $6\text{cm}$ from the fulcrum.

1. $300\text{N}$
2. $3000\text{N}$
3. $1500\text{N}$
4. $30\text{N}$

Q. As shown in Fig.7.40, the two sides of a step ladder BA and CA are 1.6 m long and hinged at A. A rope DE, 0.5 m is tied half way up. A weight 40 kg is suspended from a point F, 1.2 m from B along the ladder BA. Assuming the floor to be frictionless and neglecting the weight of the ladder, find the tension in the rope and forces exerted by the floor on the ladder. (Take g = 9.8 m/s²) (Hint: Consider the equilibrium of each side of the ladder separately.)

Q.

A uniform solid sphere of mass 1 kg and radius 10 cm is kept stationary on a rough inclined plane by fixing a highly dense particle at B. Inclination of plane is ${37}^{\circ }$ with horizontal and AB is the diameter of the sphere which is parallel to the plane, as shown in figure. Calculate

(i) Mass of the particle fixed at B

1. 3kg, 0.75

2. 4 kg, 0.75

3. 2 kg, 0.50

4. 4 kg, 0.50

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. equation of trajectory of projectilr is given by y=10x-4/7x^{2 }.the range of the projectile is, if g=10m/s^{

Q. A uniform metre stick of mass $250\text{gm}$ is suspended from the ceiling through two vertical strings of equal lengths. A small object of mass $50\text{gm}$ is placed on the stick at a distance of $70\text{cm}$ from the left end. Find the tension in both the strings. Take $g=10{\text{m/s}}^2$.

1. $T_1=1.15\text{N},T_2=2.15\text{N}$
2. $T_1=1.06\text{N},T_2=1.14\text{N}$
3. $T_1=1.40\text{N},T_2=1.60\text{N}$
4. $T_1=1.72\text{N},T_2=1.28\text{N}$

Q.

A 1kg rod of length 1m is pivoted at its centre and two masses of 5kg and 2kg and are hung from the ends as shown in figure. Find the tension in the supports to the blocks of mass 2kg and 5kg $(g=9.8\frac{m}{s^2})$

1. 27.6N, 20.6N

2. 27.6N, 29N

3. 20.6N, 29N

4. 20.6N, 20.6N

Q. A car weighs 1800 kg. The distance between its front and back axles is 1.8 m. Its centre of gravity is 1.05 m behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.

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 solid cylinder of mass m = 4 kg and radius R = 10 cm has two ropes wrapped around it, one near each end. The cylinder is held horizontally by fixing the two free ends of the cords to the hooks on the ceiling such that both the cords are exactly vertical. The cylinder is released to fall under gravity. Find the tension in the cords when they unwind and the linear acceleration of the cylinder.

1. $6.5\frac{m}{s^2},6.5N$

2. $5.5\frac{m}{s^2},6.5N$

3. $6.5\frac{m}{s^2},5.5N$

4. $5.5\frac{m}{s^2},5.5N$

Q. A light rod of length $l=1\text{m}$ is hinged at its centre and two masses of $m_1=5\text{kg}$ and $m_2=2\text{kg}$ are hung from the ends as shown in figure. Find the initial angular acceleration of the rod assuming that it was horizontal in beginning. (Take $g=10m/s^2$)

1. $8.5{\text{rad/s}}^2$
2. $4.4{\text{rad/s}}^2$
3. $3.4{\text{rad/s}}^2$
4. $5.4{\text{rad/s}}^2$

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

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

Q.

One fourth length of a uniform rod of length 2l and mass m is placed on a horizontal table and the rod is held horizontal. The rod is released from rest. Find the normal reaction on the rod as soon as the rod is released

1. $\frac{3mg}{29}$

2. $\frac{3mg}{31}$

3. $\frac{4mg}{27}$

4. $\frac{27mg}{31}$

Q.

For the toppling of the hexagon as shown in the figure, the coefficient of friction must be

1. > 0.21
2. < 0.21
3. = 0.21
4. < 0.21

Q.

Determine the point of application of net force, when forces of 20 N & 30 N are acting on the rod as shown in figure.

1. 70 cm from point A

2. 40 cm from C

3. 30 cm from D

4. 60 cm from A

Q. A body shown below is in rotational equilibrium. M is equal to ___

1. 5 Nm
2. 10 Nm
3. 15 Nm
4. 20 Nm

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. A light rod of length $l=1\text{m}$ is hinged at its centre and two masses of $m_1=5\text{kg}$ and $m_2=2\text{kg}$ are hung from the ends as shown in figure. Find the initial angular acceleration of the rod assuming that it was horizontal in beginning. (Take $g=10m/s^2$)

1. $8.5{\text{rad/s}}^2$
2. $4.4{\text{rad/s}}^2$
3. $3.4{\text{rad/s}}^2$
4. $5.4{\text{rad/s}}^2$

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. 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 1kg rod of length 1m is pivoted at its centre and two masses of 5kg and 2kg and are hung from the ends as shown in figure. Find the tension in the supports to the blocks of mass 2kg and 5kg $(g=9.8\frac{m}{s^2})$

1. 20.6N, 29N

2. 20.6N, 20.6N

3. 27.6N, 20.6N

4. 27.6N, 29N

Q.

In the system shown in figure blocks A and B have mass $m_1=2kgand{m}_2=\frac{26}{7}$ kg respectively. Pulley having moment of inertia I = 0.11 kg $m^2$ can rotate without friction about a fixed axis. Inner and outer radii of pulley are a = 10 cm and b = 15 cm respectively. B is hanging with the thread wrapped around the pulley, while A lies on a rough inclined plane.


Coefficient of friction being $\mu =\frac{\sqrt{3}}{10}$

$\begin{array}{cc}ColumnI& ColumnII\\ P.\text{Tension in the thread connecting block A}& w.26N\\ Q.\text{Tension in the thread connecting block B}& x.2\frac{m}{s^2}\\ RaccelerationofA& \\ R.\text{Acceleration of A}& y.3\frac{m}{s^2}\\ S.\text{Acceleration of B}& z.17N\end{array}$

1. P-w, Q-z, R-x, S-y

2. P-z, Q-w, R-y, S-x

3. P-z, Q-w, R-x, S-y

4. P-z, Q-z, R-y, S-y