Question

The little lion is playing in a pool, in the coast of the city of Antofagasta, with a (closed) cube of edge s and density
homogeneous ρ c. If you fully submerge the block, as the figure shows, such that your
center of mass is at a depth H. Find:

a) The mass of the block.
b) The net force on the upper face and on the face bottom of the block.
c) The thrust that the block experiences.
d) The apparent weight of the block for the child.
e) If the child drops the block, with what acceleration falls to the bottom of the pool?

Answer 1

a) The mass of the cube is simply given by $m=\rho\times V=\rho s^3$

b) The pressure under the water surface is given by

$P=\rho_w g d$

where d is the depth

The pressure acts perpendicular to the faces of the block. If the block is not too big, the pressure does not vary much and is the same in all the faces.

$F=P\times A=Ps^2$

and all forces cancel out (i.e. te forces due to the water pressure)

But to be more precise, in the given situation, the center of mass of the block is at a depth of H. Thus, the top face is at H- s/2 and the bottom face is at a depth of H+s/2 (I have chosen 0 to be the depth at the surface of the water and when going down , it is increasingly positive)

Thus, the force on the top face is
$F=Ps^2=\rho_w g (H-s/2)s^2$

and on the bottom face:

$F=Ps^2=\rho_w g (H+s/2)s^2$

c) The thrust is due to the combined effect of this force imbalance of the above part and the pull due to gravity.

The imbalance is:

$\Delta F=\rho_w g s^2((H+s/2)-(H-s/2))=\rho gs^3$

and this is acting upwards. (since force on the bottom face is greater)

The force due to gravity is mg downwards

Thus, the net force is:

$F_{net}=\rho_w gs^3-mg=\rho_w gs^3-\rho s^3g=s^3g(\rho_w-\rho)$

d) The apparent weight is just the actual weight minus the buoyant upward force due to the water.

$W_{app}=mg-\rho_w gs^3=\rho s^3g-\rho gs^3=s^3g(\rho-\rho_w)$

e) The acceleration is just :
$F=ma\implies a=\frac{F}{m}=\frac{s^3g(\rho-\rho_w)}{\rho s^3}$

Note: in all the above calculations, $\rho_w$ represents the density of water that is assumed to be constant at all depths.