Tension in string holding post

Homework Help
1

a wooden post with density 650 kg/m3 with cross section 9cm x 9cm and length 5m, hangs vertically from a string such that 3m of the pole is submerged and 2m is above the surface. What is the tension in the string

2

PK Bot’s answer:

  1. ENGINEERING CONTEXT: A wooden post hanging partly submerged must remain in net force equilibrium such that the tension in the string counteracts the difference between its weight and the buoyant lift from the displaced water. There are a number of applications in which buoyancy is used to increase or decrease tension on ropes or chains – anchored marine buoys, tethering of underwater vehicles, buoyancy modules on deepwater pipe installations, etc.

  2. PHYSICS: The dominant effects are the gravitational force (weight) of the post and the upward buoyant force resulting from the submerged volume.

  3. ASSUMPTIONS: We assume the wooden post has a uniform density, the water is quiescent with density 1000\ \mathrm{kg/m^3}, and the string’s mass is negligible; these assumptions are standard in introductory buoyancy problems and simplify the force balance.

  4. SOLUTION TECHNIQUE:
    Let the cross-sectional area be

A=0.09\times0.09=0.0081\ \mathrm{m^2}.

The total volume of the post is

V_{\mathrm{post}}=5A=5\times0.0081=0.0405\ \mathrm{m^3},

so its weight is

W=\rho_{\mathrm{wood}}gV_{\mathrm{post}}=650\times9.81\times0.0405\approx258\ \mathrm{N}\,.

The submerged volume is

V_{\mathrm{sub}}=3A=3\times0.0081=0.0243\ \mathrm{m^3},

yielding a buoyant force

F_B=\rho_{\mathrm{water}}gV_{\mathrm{sub}}=1000\times9.81\times0.0243\approx238\ \mathrm{N}\,.

For equilibrium, the tension T in the string must satisfy

T=W-F_B\approx258-238\approx20\ \mathrm{N}\,.
  1. REFLECTION: The solution, which neglects viscous and dynamic effects, indicates that the string supports approximately 20\ \mathrm{N}. It is a simple problem, but it shows that weights can be manipulated with reduced forces if buoyancy is used.

To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 4.1: Watercraft (RELEVANT)
  2. Chapter 4.3: Archimedes’ Law (RELEVANT)
  3. Chapter 4 Engineering Models: Hydrostatic Equation (RELEVANT)
  4. Chapter 3 Engineering Models: Hydrostatic Equation (RELEVANT)
  5. Chapter 3.2: Hydrostatic Equation (RELEVANT)

Disclaimer:

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