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If an object is floating (partially or fully), does the buoyancy force = the weight of the entire object?

My confusion is,

situation A: 100g object is 50% submerged and floating

situation B: 100g object is 100% submerged and floating

in both situations, wouldn't the buoyancy force = the weight of object = 1000 N? despite different volumes of water being displaced in the two situations.

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    $\begingroup$ This just means that object B has the same density as the fluid, while object A has half the density of the fluid. $\endgroup$
    – secavara
    Mar 18, 2023 at 13:19
  • $\begingroup$ ty, i figured that out after i asked ahaha $\endgroup$
    – mar01
    Mar 18, 2023 at 13:19

3 Answers 3

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That's right, the buoyancy force is equal to the weight of the object. You can figure this out simply by drawing a free-body diagram, and neglecting factors such as the density of the object. The only two forces acting on the floating object are gravity (down) and buoyancy (up). As long as the object is in equilibrium, these two forces must exactly cancel out, which shows you why the buoyancy and weight are equal - even in the case of an extremely lightweight object for which 99% floats above the surface of the water!

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Consider a bag of water floating in water. There are two forces - gravity and buoyancy force. The two are equal and opposite. You know this because a bag of water sits quietly without acceleration. So the buoyancy forces from the water outside press on the bag strongly enough to hold it up against the weight of the water pressing on the inside.

This is much like a mass sitting on a table. Somehow the table supplies an upward force equal and opposite to gravity. One difference is that people explain why pressure leads to a buoyancy force, which is often confusing. For the reaction force of a table, people don't try to explain it until more advanced courses.

Suppose you repeat this with a bag of the same size and shape full of air. The weight is much less, but the buoyancy force doesn't change. The buoyancy force is determined by the size and shape of the bag.

Repeat again with situation B. The bag is big enough to hold 100 g of water. You put a 100 g object in that is the same density as water. This means it has the same volume as 100 g of water. It just fills the bag. It floats just like it had water in it.

Repeat with situation A. This time we use a 100 g object that is half as dense as water. It has twice the volume of 100 g of water. Half the object sticks up out of the bag above the water. The buoyancy force comes from the shape of the bag, so it is strong enough to support a 100 g object. The downward force is weight of the 100 g object. So the object floats half out of the water.

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As we know from Archimedes principle, an object submerged in a non compressive fluid will experience a buoyant force vertically upright such that: $ F = - \rho g V$ where F is the buoyant force, $\rho$ is the density of the fluid and V is the volume of fluid displaced by the object, knowing this we can then study how and why an object floats or partially floats. We know that for an object to float then we need the buoyant force to be greater than the weight such that $F_b > w_b$ thus: $$ \rho_f g V > mg \\ \rho_f g V > (\rho_b V)g\\ \rho_f > \rho_b $$ From this it is easy to see that for a body to "float" inside of a fluid then we would need: $$ \rho_f = \rho_b$$ Where $\rho_f$ is the density of the fluid and $\rho_b$ is the density of the fluid, to answer your question, in situation A the buoyant force is 150% the weight of the object and in situation B the buoyant force is equal to the weight of the object, in other words: $$ \rho_A > \rho_f \\ \rho_B = \rho_f $$ I hope this gave you some insight regarding your confusion.

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