Problem — Hammer and feather in a vacuum

Problem

A hammer and a feather rest on the two pans of an equal-arm balance, in equilibrium in air. If the whole system is now placed under a vacuum bell, does the equilibrium hold? Discuss in terms of weight, Archimedes’ thrust and mass.

Solution

Step 1 — Equilibrium in air. On each pan act the weight $P = mg$ downward and the Archimedes’ thrust of the air $F_A = \rho_\text{aria}\,V\,g$ upward, where $V$ is the object’s volume. The balance compares the effective downward forces, i.e. the apparent weights:

$P_\text{app} = mg - \rho_\text{aria}\,V\,g$

At equilibrium the two pans balance: $m_\text{mart}\,g - \rho_\text{aria}V_\text{mart}\,g = m_\text{piuma}\,g - \rho_\text{aria}V_\text{piuma}\,g$.

Step 2 — The role of volumes. The feather, much less dense than the hammer, has for equal mass an enormously larger volume $V_\text{piuma}$ than $V_\text{mart}$. So the feather receives a much larger Archimedes’ thrust. For the pans to be in equilibrium in air, the hammer must have a slightly greater mass than the feather, so as to compensate for the smaller thrust it receives:

$m_\text{mart} - m_\text{piuma} = \rho_\text{aria}\,(V_\text{mart} - V_\text{piuma}) < 0 \;\Rightarrow\; m_\text{mart} > m_\text{piuma}$

Step 3 — Under vacuum. Removing the air, $\rho_\text{aria} \to 0$: both Archimedes’ thrusts vanish. Now the balance compares the true weights $mg$. Since $m_\text{mart} > m_\text{piuma}$, the hammer’s pan goes down:

$P_\text{mart} = m_\text{mart}\,g > m_\text{piuma}\,g = P_\text{piuma}$

Result.

$\ev{\text{No: in a vacuum the balance tips towards the hammer}}$

Discussion. The balance in air does not compare masses but apparent weights; the equality of apparent weights hides different masses, because the “bulky” feather was partly held up by the air’s thrust. With the air removed, that small help disappears exactly where it was greatest (on the feather), and the hammer, which is genuinely more massive, prevails.

Linked notes

Topics: Fluidostatica e fluidodinamica Concepts: Principio di Archimede · Forza peso