A bucket with mass m and capacity V is taken out of the well with water. The density of the material from

A bucket with mass m and capacity V is taken out of the well with water. The density of the material from which the bucket is made is equal to ρ, the density of water is ρw. What force must be applied to lift this bucket while it is under water and when it was pulled out of the water? Disregard the resistance of the water to the movement of the bucket.

According to Newton’s 1 law, the force F with which the bucket is pulled upward must be no less than the force with which the bucket is pulled down.

Above the surface of the water.

The bucket and the water in it are affected by gravity.
The bucket is acted upon by the force of gravity m * g, the gravity force mw * g acts on the water in the bucket, where m is the mass of the empty bucket, mw is the mass of water, g is the acceleration of gravity.

F1 = m * g + mb * g = (m + mb) * g.

Let’s find the force of gravity that acts on the water in the bucket.
mw = ρw * V, where ρw is the density of water, V is the volume of water in the bucket.

F1 = (m + ρw * V) * g.

In water.

In the water, the bucket is also acted upon by the buoyant force of Archimedes Farch, directed vertically upward.

The buoyancy force of Archimedes is determined by the formula: Farch = ρw * g * Vw. Where ρw is the density of the fluid in which the body is immersed, g is the acceleration of gravity, Vw is the volume of material from which the bucket is made.
F2 = m * g – ρv * g * Vv.

Let’s find Vv – the volume of material from which the bucket is made.
Vw = m / ρ.

F2 = m * g – ρw * g * m / ρ = m * g * (1 – ρw / ρ).