Free body diagrams

A block of mass 200 g is kept stationary on a smooth inclined plane by applying a minimum horizontal force $F=\sqrt x$ N as shown in figure.

The value of $x=$ ______.

A mass of 10 kg is suspended vertically by a rope of length 5 m from the roof. A force of 30 N is applied at the middle point of rope in horizontal direction. The angle made by upper half of the rope with vertical is $\theta=\tan^{-1}\left(x\times10^{-1}\right)$. The value of $x$ is ______.

(Given, $g=10$ m/s$^2$)

Four forces are acting at a point P in equilibrium as shown in figure. The ratio of force $F_1$ to $F_2$ is $1:x$ where $x=$ ______.

A block of mass M slides down on a rough inclined plane with constant velocity. The angle made by the incline plane with horizontal is θ. The magnitude of the contact force will be :

Two bodies of masses m$_1$ = 5 kg and m$_2$ = 3 kg are connected by a light string going over a smooth light pulley on a smooth inclined plane as shown in the figure. The system is at rest. The force exerted by the inclined plane on the body of mass m$_1$ will be :[Take g = 10 m s$^{-2}$]

A point charge $q$ of mass $m$ is suspended vertically by a string of length $l$. A point dipole of dipole moment $\vec p$ is now brought towards $q$ from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is $N\times(mgh)$, where $g$ is the acceleration due to gravity, then the value of $N$ is ______. (Note that for three coplanar forces keeping a point mass in equilibrium, $\dfrac{F}{\sin\theta}$ is the same for all forces, where $F$ is any one of the forces and $\theta$ is the angle between the other two forces)

A block of weight 100 N is suspended by copper and steel wires of same cross sectional area 0.5 cm² and, length $\sqrt3$ m and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30° and 60°, respectively. If elongation in copper wire is $(\Delta l_C)$ and elongation in steel wire is $(\Delta l_S)$, then the ratio $\dfrac{\Delta l_C}{\Delta l_S}$ is ______.

[Young's modulus for copper and steel are $1\times10^{11}\ \mathrm{N/m^2}$ and $2\times10^{11}\ \mathrm{N/m^2}$, respectively.]

A block $M$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at $O$. A transverse wave pulse (Pulse 1) of wavelength $\lambda_0$ is produced at point $O$ on the rope. The pulse takes time $T_{OA}$ to reach point $A$. If the wave pulse of wavelength $\lambda_0$ is produced at point $A$ (Pulse 2) without disturbing the position of $M$ it takes time $T_{AO}$ to reach point $O$. Which of the following options is/are correct?

Given in the figure are two blocks A and B of weight 20 N and 100 N, respectively. These are being pressed against a wall by a force F as shown. If the coefficient of friction between the blocks is 0.1 and between block B and the wall is 0.15, the frictional force applied by the wall on block B is :

Two long current carrying thin wires, both with current I, are held by insulating threads of length L and are in equilibrium as shown in the figure, with threads making an angle 'θ' with the vertical. If wires have mass λ per unit length then the value of I is :

(g = gravitational acceleration)

A block of mass $m_1=1$ kg another mass $m_2=2$ kg, are placed together (see figure) on an inclined plane with angle of inclination $\theta$. Various values of $\theta$ are given in List I. The coefficient of friction between the block $m_1$ and the plane is always zero. The coefficient of static and dynamic friction between the block $m_2$ and the plane are equal to $\mu=0.3$. In List II expressions for the friction on block $m_2$ are given. Match the correct expression of the friction in List II with the angles given in List I, and choose the correct option. The acceleration due to gravity is denoted by $g$.

[Useful information: $\tan(5.5^\circ)\approx0.1$; $\tan(11.5^\circ)\approx0.2$; $\tan(16.5^\circ)\approx0.3$]

A heavy box is to be dragged along a rough horizontal floor. To do so, person A pushes it at an angle 30° from the horizontal and requires a minimum force $F_A$, while person B pulls the box at an angle 60° from the horizontal and needs minimum force $F_B$. If the coefficient of friction between the box and the floor is $\dfrac{\sqrt3}5$, the ratio $\dfrac{F_A}{F_B}$ is :

A small block of mass of 0.1 kg lies on a fixed inclined plane $PQ$ which makes an angle $\theta$ with the horizontal. A horizontal force of 1 N acts on the block through its center of mass as shown in the figure. The block remains stationary if (take $g=10\ \mathrm{m/s^2}$)

Two particles of mass $m$ each are tied at the ends of a light string of length $2a$. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance '$a$' from the center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force $F$. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes $2x$, is