Mechanics

# Archimedes theorem

On a sunny and hot day, nothing better than a nice dip in the pool, sea or river to cool off.

When we are immersed in water we can feel a pleasant sensation of lightness in our bodies. This happens because when we are immersed in a fluid, it exerts a force that pushes us upwards.
Archimedes was the first to verify this phenomenon during a bath. After this discovery, he established the Buoyancy Theorem or Archimedes’ Theorem.

“A fluid in equilibrium acts on a body immersed in it (partially or totally), with a vertical force directed from the bottom up, called buoyancy, applied to the center of gravity of the volume of displaced fluid, whose magnitude is equal to the weight of the volume of fluid displaced. displaced fluid”.

When we dip a stone in water contained in a glass, we observe that the water level rises. According to Archimedes’ Theorem, it has an intensity equal to the weight of the volume of liquid displaced by the body, that is:

E = P L

as   L = m L . g :

E = mL . g (1)

We also know that:

mL = dL . L (2)

where:
L is the mass of the displaced liquid
L is the density of the displaced liquid
L is the volume of the displaced liquid

If we substitute equation (2) in equation (1) we have that the buoyancy can be calculated by:

E = dL . V L . g

In the diagram below we can verify the buoyancy theorem, with the aid of a balance with two equal arms.

Note that in this case the scale is in equilibrium. This indicates that the weight on the left arm of the balance is equal to the tension in the wire attached to the pan of the right arm.

P = T

Now look at the figure below:

When immersed in water, the body appears to weigh less. This happens because of the Buoyancy, that is, the liquid exerts an upward force on the body attached to the wire, causing the balance of the balance to break.

In this situation the traction of the body is:

T’ = P – E

Where:
T’ is the traction of the submerged body
P is the real weight of the body
E is the buoyancy

This new found traction can also be called the apparent weight.

ap = P – E

Therefore, the apparent weight of the body is the difference between the real weight and the buoyancy.

Where:
L is the weight of the displaced liquid.
Knowing that weight is the product of mass and gravitational acceleration.