Buoyancy Force on a Submerged Object

Buoyancy Force on a Submerged Object#

The buoyancy force arises from variations in hydrostatic pressure acting on the surfaces of a submerged object. This section derives Archimedes’ Principle from the fundamental hydrostatic equation and explains its implications.

1. Relating Buoyancy to the Hydrostatic Equation#

The pressure in a static fluid varies with depth according to the hydrostatic equation:

\[ \frac{dp}{dz} = -\rho_f g, \]

where:

  • \(p\) is the pressure,

  • \(z\) is the vertical coordinate (positive upward),

  • \(\rho_f\) is the fluid density,

  • \(g\) is the gravitational acceleration.

Integrating this equation from the surface of the fluid (\(z = 0\)) to a depth \(z\), with surface pressure \(p_0\):

\[ p = p_0 + \rho_f g z. \]

This pressure variation generates a net upward force on submerged objects, leading to the buoyancy force.

2. Derivation of Archimedes’ Principle#

Consider a submerged object of arbitrary shape fully surrounded by a fluid. The object experiences pressure forces acting on its surface.

a. Net Vertical Force#

The pressure on the bottom surface is greater than on the top surface due to the increase in pressure with depth. Let the object displace a volume of fluid \(V_d\).

  • Force on the top surface:
    At depth \(z_{\text{top}}\), pressure is:

    \[ p_{\text{top}} = p_0 + \rho_f g z_{\text{top}}. \]

    The force on the top surface (area \(A_{\text{top}}\)):

    \[ F_{\text{top}} = p_{\text{top}} A_{\text{top}}. \]

  • Force on the bottom surface:
    At depth \(z_{\text{bottom}}\), pressure is:

    \[ p_{\text{bottom}} = p_0 + \rho_f g z_{\text{bottom}}. \]

    The force on the bottom surface (area \(A_{\text{bottom}}\)):

    \[ F_{\text{bottom}} = p_{\text{bottom}} A_{\text{bottom}}. \]

  • Net upward force:
    Subtracting the forces:

    \[ F_B = F_{\text{bottom}} - F_{\text{top}} = \rho_f g (z_{\text{bottom}} - z_{\text{top}}) A_{\text{projected}}, \]

    where \(A_{\text{projected}}\) is the projected cross-sectional area.

b. Displaced Fluid Volume#

The product \((z_{\text{bottom}} - z_{\text{top}}) A_{\text{projected}}\) equals the volume of displaced fluid, \(V_d\). Thus, the buoyant force becomes:

\[ F_B = \rho_f g V_d. \]

c. Archimedes’ Principle#

This result states that the buoyant force equals the weight of the displaced fluid:

\[ F_B = \text{Weight of Displaced Fluid}. \]

3. Physical Interpretation of Buoyancy#

The buoyancy force acts upward because the fluid pressure increases with depth, creating a net upward pressure difference on the object. The magnitude of this force depends only on the volume of displaced fluid and the fluid’s density, not the properties of the object.

4. Conditions for Buoyancy#

a. Floating Objects#

If the object’s weight, \(W\), is less than the buoyant force, it floats. The equilibrium condition is:

\[ F_B = W \quad \text{or} \quad \rho_f g V_d = m_o g, \]

where \(m_o\) is the object’s mass.

b. Submerged Objects#

For fully submerged objects, the displaced fluid volume equals the object’s volume, \(V_o\). The net force determines whether the object sinks or remains neutrally buoyant:

\[ F_{\text{net}} = F_B - W. \]

5. Example Derivation#

Example: Submerged Cube
A cube with side length \(L = 0.5 \, \text{m}\) is submerged in water (\(\rho_f = 1000 \, \text{kg/m}^3\)). Find the buoyant force.

Solution:#

  1. Volume of Displaced Fluid:

    \[ V_d = L^3 = (0.5)^3 = 0.125 \, \text{m}^3. \]

  2. Buoyant Force:

    \[ F_B = \rho_f g V_d = (1000)(9.81)(0.125) = 1226.25 \, \text{N}. \]

6. Applications of Buoyancy#

a. Ship Design#

Ships float because their hulls displace enough water to create a buoyant force equal to their weight.

b. Submarines#

Submarines adjust their buoyancy by altering ballast water, controlling their submerged depth.

c. Hot Air Balloons#

In gases, the buoyant force depends on the displaced air’s weight, enabling lift.

7. Key Takeaways#

  • The buoyant force arises due to hydrostatic pressure variations in a fluid.

  • Archimedes’ Principle states that the buoyant force equals the weight of the displaced fluid.

  • Understanding buoyancy is critical in engineering applications like shipbuilding, underwater exploration, and aeronautics.

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