All springs, string and pulley shown in the figure are light. Initially, when both the springs were in their natural state, the system was released from rest. The maximum displacement of the block $\mathrm{m}$ is $x \times \left(\frac{5mg}{k}\right)$ Calculate $x\mathit{.}$

As shown in the figure, a spring is fixed at the bottom end of an inclined plane of inclination $37°$. A block of mass $4 \mathrm{kg}$ starts slipping down the incline from a point $4.6 \mathrm{m}$ away from the spring. The block compresses the spring by $40 \mathrm{cm}$, stops momentarily and then rebounds through a distance of $3 \mathrm{m}$ up the incline. If the spring constant of the spring is $\frac{{10}^{3}x}{8} \mathrm{N} {\mathrm{m}}^{-1}$, then the value of $x$ is (take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$),

In the figure shown, a spring of spring constant $K$ is fixed at one end and the other end is attached to the mass $‘m’$. The coefficient of friction between the block and the inclined plane is$‘\mathrm{\mu }’$. The block is released when the spring is in its natural length. Find the maximum speed of the block during the motion.  $(\mathrm{\theta } = 45°, \mathrm{\mu } =0.2, m= 20 \mathrm{kg}, k = 10 \mathrm{N} {\mathrm{m}}^{-1}, g = 10 \mathrm{m} {\mathrm{s}}^{-2})$

A block of mass, $m=2 \mathrm{kg}$ is pulled along a rough horizontal surface by applying a constant force at an angle, $\mathrm{\theta }={\tan }^{–1} 2$ with the horizontal as shown in the figure. The friction coefficient between the block and the surface is $\mathrm{\mu }=0.5$. If the block travels at a uniform velocity, $v=5 \mathrm{m} {\mathrm{s}}^{-1}$, then calculate the average power ($\mathrm{watt}$) of the applied force. (Take acceleration due to gravity $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.)

Potential energy of a particle of mass $m$ depends on distance $y$ from line $AB$ according to given relation, $U=\frac{K}{\sqrt{y^2+a^2}}$, where $K$ is a positive constant. A particle of mass $m$ is projected from, $y=\sqrt{3}a$ towards line $AB$ (perpendicular to it). Then, the minimum velocity so that it cannot return to its initial point is $\sqrt{\frac{K}{aNm}}$. Calculate $N$.

In the figure shown, there is no friction between $B$ and ground and $\mu =\frac{2}{3}$ between $A$ and $B$.

A body of constant mass $m=1 \mathrm{kg}$ moves under variable force $F$ as shown. If at $t=0, S=0$ and velocity of the body is $\sqrt{20} \mathrm{m} {\mathrm{s}}^{-1}$ and the force is always along direction of velocity, then choose the incorrect options 

A block of mass $m$ is attached to one end of a massless spring of spring constant $k$. The other end of the spring is fixed to a wall. The block can move on a horizontal rough surface. The coefficient of friction between the block and the surface is $\mu$. The block is released when the spring has a compression $\frac{2\mu mg}{k}$of then choose the incorrect option(s):