Two blocks of mass $2 \ kg$ and $1 \ kg$ are connected by an ideal spring on a rough surface. The spring is unstretched. The spring constant is $8 \ N/m$. The coefficient of friction is $\mu = 0.8$. Now,the $2 \ kg$ block is imparted a velocity $u$ towards the $1 \ kg$ block. Find the maximum value of velocity $u$ of the $2 \ kg$ block such that the $1 \ kg$ block never moves.

$A$ string of negligible mass going over a clamped pulley of mass $m$ supports a block of mass $M$ as shown in the figure. The force on the pulley by the clamp is given by

$A$ ball of mass $0.1\, kg$ is whirled in a horizontal circle of radius $1\, m$ by means of a string at an initial speed of $10\, rpm$. Keeping the radius constant,the tension in the string is reduced to one-quarter of its initial value. The new speed is ....... $rpm$.

Three forces acting on a body are shown in the figure. To have the resultant force only along the $y-$ direction,the magnitude of the minimum additional force needed is ........... $N$.

$A$ projectile is thrown from a point $O$ on the ground at an angle $45^{\circ}$ from the vertical and with a speed $5 \sqrt{2} \text{ m/s}$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground,$0.5 \text{ s}$ after the splitting. The other part,$t$ seconds after the splitting,falls to the ground at a distance $x$ meters from the point $O$. The acceleration due to gravity $g = 10 \text{ m/s}^2$.
$(1)$ The value of $t$ is. . . . . .
$(2)$ The value of $x$ is. . . . .

In the figure,a ball of mass $m$ is tied with two strings of equal length as shown. If the rod is rotated with angular velocity $\omega$,then