Newton's second law states that the resultant of the forces acting on the sphere is equal to the product between its mass m and its acceleration a:
[tex]\sum F = ma[/tex]
There are two forces acting on the sphere: its weight W, directed downward, and the buoyancy B, directed upward. So Newton's second law becomes
[tex]W-B = ma[/tex] (1)
We already know the weight of the sphere: [tex]W=mg=400 N[/tex], from which we can also find the mass of the sphere:
[tex]m= \frac{W}{g}= \frac{400 N}{9.81 m/s^2}=40.8 kg [/tex]
The buoyancy is equal to the mass of the water displaced:
[tex]B=\rho_W V g[/tex]
where
[tex]\rho_W = 1000 kg/m^3[/tex] is the water density
V is the volume of water displaced, which corresponds to the volume of the sphere, since the sphere is underwater
g is the gravitational acceleration
The radius of the sphere is
[tex]r= \frac{40 cm}{2}=20 cm=0.20 m [/tex], so its volume is
[tex]V= \frac{4}{3} \pi r^3 = \frac{4}{3}\pi (0.20 m)^3 = 3.55 \cdot 10^{-2} m^3 [/tex]
So now we can rewrite Newton's second law (1) as
[tex]mg-\rho_W V g = ma[/tex]
and solve it to find the acceleration of the sphere, a:
[tex]a=g- \frac{\rho_W V g}{m}=9.81 m/s^2 - \frac{(1000 kg/m^3)(3.55 \cdot 10^{-2} m^3)(9.81 m/s^2)}{40.8 kg} =1.76 m/s^2 [/tex]
and this acceleration is directed downward, since it has the same sign of g.
