Understanding Buoyancy and Submerged Volume

This problem asks us to compare the fraction of a sphere's volume submerged when floating in water under two different conditions: on Earth under gravity and on an accelerating spaceship in outer space.

The key principle here is that for an object to float, the buoyant force acting on it must be equal in magnitude to its weight. The buoyant force is the weight of the fluid displaced by the submerged part of the object.

Let V be the total volume of the sphere, \(\rho_s\) be the density of the sphere, and \(\rho_w\) be the density of water. The sphere floats because its density is less than the density of water (\(\rho_s < \rho_w\)).

Case 1: Sphere Floating on Earth

On Earth, the acceleration due to gravity is g.

• Weight of the sphere (W_Earth) = Mass × Acceleration = \(\rho_s V g\)
• Volume submerged = f₁V
• Volume of displaced water = f₁V
• Weight of displaced water (Buoyant Force, B_Earth) = Density of water × Volume of displaced water × Acceleration = \(\rho_w (f_1 V) g\)

Since the sphere is floating, the buoyant force equals its weight:

B_Earth = W_Earth

\(\rho_w (f_1 V) g = \rho_s V g\)

We can cancel out the volume V (since V > 0) and the acceleration g (since g > 0) from both sides:

\(\rho_w f_1 = \rho_s\)

Thus, the fraction submerged on Earth is:

\(f_1 = \frac{\rho_s}{\rho_w}\)

Case 2: Sphere Floating on an Accelerating Spaceship

On the spaceship in outer space, there is an acceleration a (\(a < g\)). This acceleration acts as the effective gravitational acceleration within the spaceship's frame of reference, causing the water to settle and objects to experience weight and buoyancy relative to this acceleration.

• Effective acceleration = a
• Weight of the sphere (W_Spaceship) = Mass × Effective acceleration = \(\rho_s V a\)
• Volume submerged = f₂V
• Volume of displaced water = f₂V
• Weight of displaced water (Buoyant Force, B_Spaceship) = Density of water × Volume of displaced water × Effective acceleration = \(\rho_w (f_2 V) a\)

Since the sphere is floating on the spaceship, the buoyant force equals its weight:

B_Spaceship = W_Spaceship

\(\rho_w (f_2 V) a = \rho_s V a\)

We can cancel out the volume V (since V > 0) and the effective acceleration a (since a > 0) from both sides:

\(\rho_w f_2 = \rho_s\)

Thus, the fraction submerged on the spaceship is:

\(f_2 = \frac{\rho_s}{\rho_w}\)

Comparing the Submerged Fractions

From Case 1, we found \(f_1 = \frac{\rho_s}{\rho_w}\).

From Case 2, we found \(f_2 = \frac{\rho_s}{\rho_w}\).

Therefore, \(f_1 = f_2\).

The fraction of the volume submerged when an object floats in a fluid depends only on the ratio of the object's density to the fluid's density. It does not depend on the magnitude of the uniform acceleration (gravity or effective acceleration) acting on the system, as long as that acceleration is non-zero and uniform throughout the fluid and the object.

Conclusion

The submerged volume on the spaceship (\(f_2V\)) is the same fraction of the total volume as it is on Earth (\(f_1V\)). Thus, \(f_2 = f_1\).

The value of acceleration (g or a) cancels out from the equation when determining the fraction of volume submerged.

Revision Table: Buoyancy Concepts

Additional Information: Buoyancy in Different Frames

Understanding buoyancy in different reference frames, like an accelerating spaceship, is a good way to solidify the concepts of inertial frames and effective forces.

• In the accelerating spaceship's frame, the acceleration 'a' acts like a uniform gravitational field. The physics principles, including Archimedes' principle, apply in this non-inertial frame if we include this effective acceleration in calculations of weight and buoyant force.
• Both the weight of the sphere and the weight of the displaced water are directly proportional to the effective acceleration. When setting these two forces equal for the floating condition, the acceleration term cancels out, meaning the submerged fraction is independent of the acceleration's magnitude (as long as it's uniform and non-zero).
• This is analogous to how your weight and the buoyant force on you in an elevator accelerating upwards or downwards both change proportionally, but whether you float or sink in a fluid inside the elevator would still depend on your density relative to the fluid's density, not the elevator's acceleration.
• The density ratio (\(\rho_s / \rho_w\)) is an intrinsic property of the sphere and the water, independent of the external acceleration field.