An astronaut, in a space shuttle, orbiting close to the earth's surface (take $g= 10~\text{m/s}^2$), suddenly fires his engines so as to give him a forward acceleration of $\dfrac{3g}{4}$ along the direction of his motion. At that instant, his apparent weight is:

The mass $m$ can slide along a smooth radial groove in a horizontal turntable; at the other end is attached a spring- so that the mass $m$ presses against the spring as it moves 'outward'. The free-end ($A$) of the spring is at a distance $b$ from the centre and the spring constant is $k.$ If the turntable is rotated with an angular speed $\omega$, and the mass $m$ is in 'equilibrium' with the spring compressed, then the compression is:

       

A ball of mass $m$ falls from a height $h$ onto the ground and rebounds to a height $\dfrac{h}{4}$. The impulse on the ball from the ground has the magnitude:

A wire connects two blocks of masses $M$, $2M$; both lying on a smooth horizontal plane.

When a force $F_1$ is applied to $2M$ as shown in figure 1, the wire just breaks. On the other hand, when $F_2$ is applied to $M$ as shown in figure 2,

the wire just breaks. Assume that the mass of the wire is negligible. Then:

The two blocks are at rest on a smooth horizontal plane and are connected by strings passing over a smooth light pulley as shown. The strings are vertical while the force $F,$ applied to the pulley is vertical. For what minimum value of $F$ will the $2$ kg block be lifted off?
($g=10$ m/s²)

The two blocks $A$ and $B$ are placed on a smooth horizontal plane, with the string initially just taut. Forces are applied as shown. The tension in the string is:
        

A block of mass $m$ is placed on a smooth inclined surface of a wedge of mass $M$. The system is placed on a smooth horizontal table and released.

Then: