$A$ particle moves from $x = 0$ to $x = x_1$ under the influence of a force $F = Cx$. Find the work done in this process.

$A$ body of $0.5 \,kg$ moves along the positive $X$-axis under the influence of a varying force $F$ (in newton) as shown below. If the speed of the object at $x=4 \,m$ is $3.16 \,ms^{-1}$,then its speed at $x=8 \,m$ is ................. $\,ms^{-1}$.

$A$ body of mass $8\,kg$ is moved by a force $F = 3x\,N$,where $x$ is the distance covered. The initial position is $x = 2\,m$ and the final position is $x = 10\,m$. The initial speed is $0.0\,m/s$. The final speed is ........... $m/s$.

$A$ particle moves along the $x$-axis from $x = 0$ to $x = 10 \ m$ under the influence of a force $\vec{F}(x) = (3x^2 - 2x + 7) \hat{i} \ N$. The work done during this displacement is ........ $J$.

$A$ block of mass $m=1 \; kg$ moving on a horizontal surface with speed $v_{i}=2 \; m \; s^{-1}$ enters a rough patch ranging from $x=0.10 \; m$ to $x=2.01 \; m$. The retarding force $F$ on the block in this range is inversely proportional to $x$ over this range,$F_{r} = -k/x$ for $0.1 < x < 2.01 \; m$,and $F_{r} = 0$ for $x < 0.1 \; m$ and $x > 2.01 \; m$,where $k=0.5 \; J$. What is the final kinetic energy $K_{f}$ and speed $v_{f}$ of the block as it crosses this patch?

What does the area enclosed by the $F-x$ graph represent? (where $F$ is the external force and $x$ is the change in the length of the spring)