In the case of ring, you have to consider linear expansion, not the areal expansion. Similarly in the care of sphere you have to consider volume expansion.
When the block in equilibrium, the net pressure on the block is zero.
Since the density of the block is more than the density of liquid, it will submerge completely in the liquid and partly in the water.
Now net upward pressure on the block = pressure due to liquid + pressure due to water.
The passenger can pull the projectile into the vehicle, when his velocity is equal the velocity of the projectile. That gives you one equation. Difference between the distances traveled by the projectile and the vehicle is given. That gives you the second equation. Solving these two equations...
Same tension T is acting on m and 2m. So the displacement in m is greater then 2m.
The lengths of the vertical segments must be the same. Hence at any instant, the total length of the string L = x1 + x2 + 2x.Taking derivative, you can find the relation between the accelerations of m, 2m and...