A block starts moving up an inclined plane of inclination \(30^\circ\) with an initial velocity of \(v_0\). It comes back to its initial position with velocity \(\frac{v_0}{2}\). The value of the coefficient of kinetic friction between the block and the inclined plane is close to \(\frac{I}{1000}\). The nearest integer to \(I\) is:
1. \(210\)
2. \(346\)
3. \(972\)
4. \(100\)

A boy pushes a box of mass \(2\) kg with a force \(\vec{F}=(20\hat{i}+10 \hat{j})~\text{N}\) on a frictionless surface. If the box was initially at rest, then the displacement along the \(\mathrm{x}\)-axis after \(10\) s is:
1. \(100\) m
2. \(300\) m
3. \(500\) m
4. \(700\) m

As shown in the figure, a block of mass \(\sqrt{3}\) kg is kept on a horizontal rough surface of coefficient of friction \(\dfrac{1}{3\sqrt{3}}\). The critical force to be applied on the vertical surface as shown at an angle \(60^\circ\) with horizontal such that it does not move, will be \(3x\). The value of \(x\) will be:
\((g=10 \mathrm{~m} / \mathrm{s}^2 ; \sin 60^{\circ}=\frac{\sqrt{3}}{2} ; \cos 60^{\circ}=\frac{1}{2})\)
$$

     
1. \(1.23\)
2. \(4.5\)
3. \(3.33\)
4. \(9.24\)