The Complete Equilibrium

Q.

A first order reaction has a rate constant $1.15\times {10}^{-3}{\mathrm{s}}^{-1}$. How long will 5 g of this reactant take to reduce to 3 g?

Q. For rotational equilibrium of a rigid body, it is essential that?

1. Net torque about points lying on axis of rotation must be zero.
2. Net torque about points lying on the body must be zero
3. Net torque about points lying outside the body must be zero
4. All of these

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. 60 cm from $A$

4. 30 cm from $D$

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 the mass of the particle fixed at B and the minimum coefficient of fiction.

1. 3kg, 0.75

2. 4 kg, 0.75

3. 4 kg, 0.50

4. 2 kg, 0.50

Q. A rod of uniform mass is tightly hinged at its centre. Two force of equal magnitude and in same direction are applied on the body at its rear ends. The body will be in?

1. Only translational equilibrium
2. Only rotational equilibrium
3. Rotational and translational equilibrium
4. Neither rotational nor translational equilibrium

Q. Three identical bodies (each mass $M$) are placed at vertices of an equilateral triangle of arm $L,$ keeping the triangle as such, by what angular speed the bodies should be rotated in their gravitational fields, so that the triangle moves along circumference of circular orbit.

1. $\sqrt{\frac{3GM}{L^3}}$
2. $3\sqrt{\frac{GM}{L^3}}$
3. $\sqrt{\frac{GM}{L^3}}$
4. $\sqrt{\frac{GM}{3L^3}}$

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

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

4. $5.5\frac{m}{s^2},6.5N$

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

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

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

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

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

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

Q. Suppose the resulting torque on a body is (i) zero (ii) not zero. What is the effect of the acting torques on the body in the two cases?

Q.

An extensible string is woung over a rough pulley of mass $m_1$ and radius R; and a cylinder of mass $m_2$ and radius R such that as the cylinder rolls down the string un-wounds over the pulley as well the cylinder. Find the linear acceleration of the cylinder.

1. $\left[\frac{2(M_1+M_2)}{3m_1+M_2}\right]g$

2. $\left[\frac{M_1+M_2}{3m_1+2M_2}\right]g$

3. $\left[\frac{2(M_1+M_2)}{2m_1+3M_2}\right]g$

4. $\left[\frac{2(M_1+M_2)}{3m_1+2M_2}\right]g$

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. $\frac{{(1+{\mu }_k)}^2{\omega }_0^{2}R}{8\pi {\mu }_k(1+{\mu }_k)g}$
2. $\frac{(1+{\mu }_k^2){\omega }_0^{2}R}{8\pi {\mu }_k(1+{\mu }_k)g}$

3. $\frac{(1+{\mu }_k^2){\omega }_0^{2}R}{8\pi {\mu }_k(1+{\mu }_k^2)6g}$

4. $\frac{(1+{\mu }_k^2){\omega }_0^{2}R}{4\pi {\mu }_k(1+{\mu }_k)g}$