Proving Archimedes' Principle for Sphere & Cylinder

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In summary, Archimedes's principle states that the buoyant force on any object submerged in a fluid is equal to the weight of the fluid displaced by the object. This can be proven by integrating the fluid pressure over the surface area or by using a simpler argument that works for any shape. Archimedes himself did not know calculus, but it can be used to calculate the buoyant force on a large object in a non-uniform gravitational field.
  • #1
sadhu
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can anyone prove Archimedes's principle for a sphere and a cylindrical vessel of same volume
, and prove that the forces are same in both cases.



I mean buoyant forces.....
 
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  • #2
Well, you could just crank it out--integrate the fluid pressure over the surface area. Sounds like a good exercise.

But you can also use an argument that requires no integration and works for any shape. Imagine any object submerged in the fluid. Now replace that object by an equal volume of fluid. Since the fluid is in hydrostatic equilibrium, the net force of the surrounding fluid on that imaginary "object" must equal the weight of the fluid contained within its boundary. That's Archimedes's principle. Done!
 
  • #3
I prefer to use that integration idea

but on calculation it is getting to complicated to solve , however somehow when i solved it
, I just got it wrong...i think

can anyone show me its integration...pleazzzz
 
  • #4
P = dgh = dg(-y), where I’m writing d for the density of the fluid, and choosing +y in the upward direction, and g is a scalar, and g=-gj.

Buoyant force B = total force on submerged body = Surface Integral Pda = Volume integral [(grad P)dV].

Now, grad P = grad (-dgy) = -dgj

So, B = the vol integral = -dgVj= -dV*gj = weight of fluid displaced, acting upward.

p.s. Archimedes didn't know calculus. Anyway, now you can try your hand on calculating the buoyant force on a really big object floating in the ocean by calculus. I mean, in a non-uniform g field.
 
Last edited:

1. What is Archimedes' Principle?

Archimedes' Principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

2. How can Archimedes' Principle be applied to spheres and cylinders?

For spheres and cylinders, Archimedes' Principle can be used to determine the volume and density of the object by measuring the weight of the displaced fluid.

3. What materials and equipment are needed to prove Archimedes' Principle for spheres and cylinders?

To prove Archimedes' Principle for spheres and cylinders, you will need a scale, a graduated cylinder, water, and the spheres and cylinders you want to test.

4. What is the procedure for proving Archimedes' Principle for spheres and cylinders?

The procedure involves weighing the object in air, then submerging it in water and measuring the weight of the displaced water. The difference between the two weights is the weight of the displaced water, which can be used to calculate the volume and density of the object.

5. What are the potential sources of error in proving Archimedes' Principle for spheres and cylinders?

Some potential sources of error include air bubbles on the surface of the object, which can affect its weight in water, and inaccuracies in measuring the volume of the displaced water. It is important to ensure that the object is completely submerged and to take multiple measurements to reduce error.

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