For the given system,find the value of ${\theta _1}$ in degrees.

Two beads connected by a massless inextensible string are placed over a fixed ring as shown in the figure. The mass of each bead is $m$,and there is no friction between bead $B$ and the ring. Find the minimum value of the coefficient of friction between bead $A$ and the ring so that the system remains in equilibrium. ($C$ is the center of the ring,and the line $AC$ is vertical.)

$A$ block of mass $M = 10\,kg$ rests on a horizontal table. The coefficient of friction between the block and the table is $\mu = 0.05.$ When hit by a bullet of mass $m = 50\,g$ moving with speed $v,$ which gets embedded in it,the block moves and comes to a stop after moving a distance of $2\,m$ on the table. If a freely falling object were to acquire speed $\frac{v}{10}$ after being dropped from height $H,$ then neglecting energy losses and taking $g = 10\,m/s^2,$ the value of $H$ is close to ................. $km$.

The masses of blocks $A$ and $B$ are $100 \, kg$ and $200 \, kg$ respectively. The coefficient of friction between $A$ and $B$ is $0.2$,and the coefficient of friction between $B$ and the ground is $0.3$. What force $F$ (in $N$) must be applied to block $B$ to make it move?

$A$ truck of mass $2800 \ kg$ is moving at a speed of $15 \ m/s$. $A$ frictional force of $500 \ N$ and a forward force of $1200 \ N$ act on it. The distance covered by the truck in $10 \ s$ is .......... $m$.

$A$ block of mass $2M$ is attached to a massless spring with spring-constant $k$. This block is connected to two other blocks of masses $M$ and $2M$ using two massless pulleys and strings. The accelerations of the blocks are $a_1, a_2$ and $a_3$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $x_0$. Which of the following option$(s)$ is/are correct? [$g$ is the acceleration due to gravity. Neglect friction]