Buoyant force on a submerged body

In summary, a hemispherical portion of a cylinder is removed and the remaining cylinder is suspended by a string in a liquid of density ρ. The force on the bottom of the cylinder by the liquid can be found by balancing the weight of the cylinder with the force on the top and bottom surfaces. Using Archimedes' principle, the buoyant force is equal to the weight of the liquid displaced, and the force on the upper surface is downward while the force on the lower surface is upward. The net buoyant force is equal to the difference between these two forces, or ρghπr2 - F, and this must be equal to the weight of the cylinder, Mg. Solving for F, we get the correct
  • #1
Vinita
23
0

Homework Statement


A hemispherical portion of radius R is removed from the bottom of a cylinder of radius R. The volume of the remaining cylinder is V and its mass is M. It is suspended by a string in a liquid of density ρ where it stays vertical. The upper surface of the cylinder is at a depth h below the liquid surface. The force on the bottom of the cylinder by the liquid is?
UPHOTO_20180329_092040.jpg


Homework Equations


Force = Pressure x Area

The Attempt at a Solution


Since the object is in equilibrium, I balanced the 3 forces acting on it.
Weight of the body = Force on the top of the cylinder - Force on the bottom of the cylinder
⇒Mg = ρgh(πR²) - F (ignoring atmospheric pressure)
⇒F = πR²hρg - Mg
But the answer is F = ρg( V + πR²h)
What did I do wrong?
 

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  • #2
Vinita said:

Homework Statement


A hemispherical portion of radius R is removed from the bottom of a cylinder of radius R. The volume of the remaining cylinder is V and its mass is M. It is suspended by a string in a liquid of density ρ where it stays vertical. The upper surface of the cylinder is at a depth h below the liquid surface. The force on the bottom of the cylinder by the liquid is?
View attachment 222955

Homework Equations


Force = Pressure x Area

The Attempt at a Solution


Since the object is in equilibrium, I balanced the 3 forces acting on it.
Weight of the body = Force on the top of the cylinder - Force on the bottom of the cylinder
⇒Mg = ρgh(πR²) - F (ignoring atmospheric pressure)
⇒F = πR²hρg - Mg
But the answer is F = ρg( V + πR²h)
What did I do wrong?

There is the force of the string you omitted.
 
  • #3
Hint: think about the forces on the three different surfaces and what they must add up to vectorially.
 
  • #4
ehild said:
There is the force of the string you omitted.
Then the equation will become
T - Mg = ρgh(πr²) - F
I have 2 variables now. How can I solved this equation?
Can I assume T ≈ Mg because the string remains taut?
 
  • #5
Vinita said:
Then the equation will become
T - Mg = ρgh(πr²) - F
I have 2 variables now. How can I solved this equation?
Can I assume T ≈ Mg because the string remains taut?
No need to get tangled up with the string. See my hint in post #3 and remember Archimedes.
 
  • #6
haruspex said:
Hint: think about the forces on the three different surfaces and what they must add up to vectorially.
3 different surfaces of the cylinder?
I tried.
UPHOTO_20180329_112130.jpg

Here the forces on the sides cancel each other.
F''cosθ cancel each other.
According to Archimedes, buoyant force= weight of liquid displaced
= Vρg
⇒ρgh(πr²) - F = Vρg
F = ρg(πr²h - V)
Which is still not correct.
 

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  • #7
Vinita said:
Which is still not correct.
True, but now it is down to a sign error. Think very carefully which force(s) you might have ascribed the wrong sign to in the equation.
 
  • #8
haruspex said:
True, but now it is down to a sign error. Think very carefully which force(s) you might have ascribed the wrong sign to in the equation.
It must be the force on the upper surface, according to the answer.
But why? The water above the object will exert a downward force on the object. Because the force should be perpendicular to the surface.
The water below the object will exert a force upward for the same reason.
 
  • #9
Vinita said:
It must be the force on the upper surface, according to the answer.
But why? The water above the object will exert a downward force on the object. Because the force should be perpendicular to the surface.
The water below the object will exert a force upward for the same reason.
What is the direction of the buoyant force? Upward or downward?
 
  • #10
ehild said:
What is the direction of the buoyant force? Upward or downward?
Upward.
The direction of net buoyant force is upward.
_20180329_124244.jpg

I found this image which relates to what I am confused about.
The force on the upper surface (say F1) is downward, but weaker due to lesser height. The force on lower surface (say F2) is stronger and upward.
Thus net force = Buoyant force = F2 - F1
Why can't we apply the same thing here as well?
 

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  • #11
Vinita said:
Upward.
The direction of net buoyant force is upward.View attachment 222965
I found this image which relates to what I am confused about.
The force on the upper surface (say F1) is downward, but weaker due to lesser height. The force on lower surface (say F2) is stronger and upward.
Thus net force = Buoyant force = F2 - F1
Why can't we apply the same thing here as well?
You can, but you didn't. Substitute F for F2 and ρghπr2 for F1 etc. in that.
 
  • #12
haruspex said:
You can, but you didn't. Substitute F for F2 and ρghπr2 for F1 etc. in that.
F - ρgh(πr²) = Vρg
Okay. I understood. It was a stupid mistake. Thanks for your assistance.:smile:
 

FAQ: Buoyant force on a submerged body

What is buoyancy?

Buoyancy is the upward force exerted by a fluid on an object that is submerged in the fluid. It is caused by the difference in pressure between the top and bottom of the object.

How is buoyancy calculated?

Buoyancy is calculated using Archimedes' principle, which states that the buoyant force on an object is equal to the weight of the fluid that the object displaces. This can be calculated by multiplying the density of the fluid by the volume of the displaced fluid.

How does the shape of an object affect its buoyancy?

The shape of an object does not affect the buoyancy force itself, but it does affect the object's ability to displace fluid. A more streamlined or hollow object will displace more fluid and therefore experience a greater buoyant force.

Does the density of the object affect its buoyancy?

Yes, the density of an object affects its buoyancy. An object with a higher density than the fluid it is submerged in will sink, while an object with a lower density will float. This is because the object's weight is greater or less than the weight of the fluid it displaces.

Why do some objects float while others sink?

Objects float when the buoyant force acting on them is greater than their weight. This is typically the case for objects with a lower density than the fluid they are submerged in. Objects sink when the buoyant force is less than their weight, which is usually the case for objects with a higher density than the fluid.

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