Calculate the buoyant force exerted by the water on the sphere

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Homework Help Overview

The discussion revolves around calculating the buoyant force on a hollow plastic sphere submerged in freshwater, determining its mass, and finding the fraction of the sphere's volume that remains submerged when it reaches equilibrium after the supporting cord is broken.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the calculation of buoyant force using the formula involving density, volume, and gravity. There are attempts to relate the buoyant force to the mass of the sphere and the tension in the cord. Questions arise about expressing the submerged volume as a percentage and the relationship between the mass of the sphere and the buoyant force.

Discussion Status

Some participants have provided calculations and reasoning regarding the buoyant force and mass of the sphere. Guidance has been offered on how to relate the submerged volume to the total volume of the sphere, indicating a productive direction in the discussion.

Contextual Notes

There is an emphasis on using the correct relationships between buoyant force, mass, and volume, with participants questioning the assumptions made in their calculations. The problem context includes specific values for density and gravitational acceleration.

MAPgirl23
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A hollow, plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.700 M^3 and the tension in the cord is 760 N.

Calculate the buoyant force exerted by the water on the sphere. Take the density of water to be 1000 kg/m^3 and the free fall acceleration to be 9.80 m/s^2.
**for this I set up buoyant = density *Volume*gravity = 6860 N

What is the mass of the sphere? Take the density of water to be 1000kg/m^3 and the free fall acceleration to be 9.80m/s^2 .
**here I used Buoyant = mg+T and solved for m, so answer for m= 622 kg

The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged? Express your answer as a percentage.
**For the sphere to be at rest, mg = B. There is no tension anymore, since the cord was broken. I express B as a function of the volume of the sphere that is still submerged:

mg = B --> rho*V*g = 0.7 how do I express it as a percentage? I tried 70%
 
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MAPgirl23 said:
A hollow, plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.700 M^3 and the tension in the cord is 760 N.

Calculate the buoyant force exerted by the water on the sphere. Take the density of water to be 1000 kg/m^3 and the free fall acceleration to be 9.80 m/s^2.
**for this I set up buoyant = density *Volume*gravity = 6860 N

What is the mass of the sphere? Take the density of water to be 1000kg/m^3 and the free fall acceleration to be 9.80m/s^2 .
**here I used Buoyant = mg+T and solved for m, so answer for m= 622 kg

The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged? Express your answer as a percentage.
**For the sphere to be at rest, mg = B. There is no tension anymore, since the cord was broken. I express B as a function of the volume of the sphere that is still submerged:

mg = B --> rho*V*g = 0.7 how do I express it as a percentage? I tried 70%

The first two look good. The mg on the left side of your last equation involves the mass of the sphere you already calculated. The buoyant force will be rho*V*g, but rho is the density of water and V is the volume of water displaced. This does not equal .700. Use the mg = B part to find B and use B = rho*V*g to find V. The ratio of that V (volume of displaced water) to the volume of the sphere is the fraction submerged.
 
622 kg * 9.8 = 6095.6 N

6095.6N / (1000 * 9.8) = 0.622

0.622 / 0.700 = 88.9% is this correct?
 
MAPgirl23 said:
622 kg * 9.8 = 6095.6 N

6095.6N / (1000 * 9.8) = 0.622

0.622 / 0.700 = 88.9% is this correct?

Looks good to me.
 

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