Exam problem

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A sphere, of radius R, is partially immersed, to depth d, in a liquid of specific gravity SG. Obtain an algebraic expression for the buoyancy force acting on the sphere as a function of submersion depth d.

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This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:

  1. ENGINEERING CONTEXT:
    This analysis is essential when designing buoyant devices such as floats, buoys, or partially submerged structures that rely on controlling the displaced volume for stability and lift. Once you complete this problem, you can apply the result to design devices where precise buoyant force control is needed.

  2. PHYSICS:
    The dominant effect is the hydrostatic pressure distribution in a quiescent fluid, which by Archimedes’ principle produces an upward force equal to the weight of the displaced fluid.

  3. ASSUMPTIONS:
    We assume the fluid is at rest and that the submerged portion of the sphere forms an ideal spherical cap of depth d; this is reasonable as long as the free-surface disturbances and dynamic effects are negligible.

  4. SOLUTION TECHNIQUE:
    The buoyancy force is given by

F_b=\rho_f\,g\,V_\mathrm{sub},

where the fluid density is

\rho_f = SG\,\rho_\mathrm{water}.

For a sphere of radius R partially immersed to a depth d, the volume of the submerged spherical cap is

V_\mathrm{sub}=\frac{\pi d^2}{3}\left(3R-d\right).

Thus, the buoyancy force becomes

F_b=SG\,\rho_\mathrm{water}\,g\,\frac{\pi d^2}{3}\left(3R-d\right).
  1. REFLECTION:
    This analytical expression utilizes Archimedes’ principle by replacing the vector surface integral with a scalar volume calculation, which is both efficient and intuitive. The result is accurate for static fluids and forms a basis for predicting the buoyant response of spherical objects in engineering applications.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 4.3: Archimedes’ Law Part 1 (HIGHLY RELEVANT)
  2. Chapter 4.1: Watercraft Part 1 (HIGHLY RELEVANT)
  3. Chapter 4.1: Watercraft Part 2 (HIGHLY RELEVANT)
  4. Chapter 4 Engineering Models: Hydrostatic Equation (HIGHLY RELEVANT)
  5. Chapter 3 Engineering Models: Hydrostatic Equation (HIGHLY RELEVANT)

Disclaimer:

See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.

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the only fluids concept is archimedes principle. The rest is calculating the volume of a spherical cap by integration or by googling/chatbotting the formula