14-5 Archimedes’ Principle

14-5 Archimedes’ Principle#

Prompts

  • State Archimedes’ principle. What is the “fluid displaced” and why does the buoyant force equal its weight?

  • A stone sinks and wood floats in water. Compare the magnitudes of buoyant force and gravitational force for each. What determines sink vs float?

  • For a floating body, how is the submerged volume related to the body’s mass and the fluid density?

  • Define apparent weight. How does it differ from actual weight when a body is submerged?

  • You hold a rock underwater. Does the scale reading (apparent weight) increase, decrease, or stay the same compared to in air?

Lecture Notes#

Overview#

  • Buoyant force: When a body is submerged (fully or partially), the fluid pushes upward with a force equal to the weight of the fluid displaced.

  • Energy view: The buoyant force arises from pressure (energy density) differences in the fluid; 14-7 unifies pressure with kinetic and gravitational energy densities.

  • Sink vs float: \(F_b > F_g\) → rises; \(F_b < F_g\) → sinks; \(F_b = F_g\) → floats (equilibrium).

  • Apparent weight: What you “feel” when submerged; actual weight minus buoyant force.

Archimedes’ principle#

Archimedes’ principle

When a body is fully or partially submerged in a fluid, the fluid exerts an upward buoyant force of magnitude

(25)#\[ F_b = m_f g \]

where \(m_f\) is the mass of the fluid displaced by the body (the fluid that would occupy the space taken by the submerged part).

Physical picture: Imagine replacing the submerged body with a sack of fluid of the same shape. That sack would be in equilibrium—its weight \(m_f g\) is balanced by the pressure forces from the surrounding fluid. Those same pressure forces act on the body. So the buoyant force equals the weight of the displaced fluid. The pressure field (energy density) in the fluid produces the net upward force.

Note

The buoyant force does not depend on the body’s density or composition—only on the volume submerged and the fluid density: \(F_b = \rho_f V_{\text{sub}} g\).

Sink, float, or rise?#

Condition

Result

\(F_b > F_g\) (\(\rho_{\text{body}} < \rho_f\))

Body rises (e.g., wood in water)

\(F_b < F_g\) (\(\rho_{\text{body}} > \rho_f\))

Body sinks (e.g., stone in water)

\(F_b = F_g\)

Floats in equilibrium

Floating bodies#

When a body floats, it is in static equilibrium: \(F_b = F_g\). So

(26)#\[ m_f g = m g \quad \Rightarrow \quad m_f = m \]

The mass of fluid displaced equals the mass of the body. Equivalently, \(\rho_f V_{\text{sub}} = \rho_{\text{body}} V_{\text{body}}\), so

(27)#\[ \frac{V_{\text{sub}}}{V_{\text{body}}} = \frac{\rho_{\text{body}}}{\rho_f} \]

A denser body sits lower (more submerged); a less dense body rides higher.

Poll: Rock in boat

A boat floats in a pond. You throw a rock from the boat into the water; the rock sinks. After the rock is at the bottom, does the water level in the pond (relative to the shore) go up, down, or stay the same?

  • (A) Up

  • (B) Down

  • (C) Same

Apparent weight#

Apparent weight: the weight you “feel” when a buoyant force acts—e.g., when you hold an object underwater or stand on a scale in a pool:

(28)#\[ W_{\text{app}} = W - F_b = mg - F_b \]

  • In air: \(F_b\) is small; \(W_{\text{app}} \approx W\).

  • Submerged in liquid: \(F_b\) is significant; \(W_{\text{app}} < W\) (object feels lighter).

Poll: Submerged rock

You hold a rock underwater. Compared to holding it in air, the force you must exert to support it (keep it from accelerating) is:

  • (A) Greater

  • (B) The same

  • (C) Less

  • (D) Zero—the rock floats

Summary#

  • Archimedes’ principle: \(F_b = m_f g\), where \(m_f\) is mass of fluid displaced

  • Sink: \(F_b < F_g\). Float: \(F_b = F_g\); \(V_{\text{sub}}/V_{\text{body}} = \rho_{\text{body}}/\rho_f\)

  • Apparent weight: \(W_{\text{app}} = W - F_b\)