Kvant Physics Problem 22

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Verdicts: SKIP + SKIP
Solve time: 1m11s
Source on kvant.digital

Problem

Two blacksmiths are working a piece of iron. First, they place it on an anvil and strike it with hammers alternately; then they suspend it from the ceiling and strike it simultaneously from different sides. The force of each blacksmith’s blow is the same in both cases. In which case will the piece of iron heat up more from a single blow?

A. M. Budker

Setup and Assumptions

A piece of iron of mass $M$ is acted upon by two blacksmiths. Each blow delivers the same impulse magnitude $J$ to the iron through a hammer. The duration of each impact is short compared with the mechanical relaxation time of the iron, so each blow can be treated as an impulsive interaction.

In the first configuration, the blacksmiths strike the iron alternately while it rests on a rigid anvil. After each impact the iron can exchange momentum with the anvil and come to rest before the next blow.

In the second configuration, the iron is suspended freely and the blacksmiths strike it simultaneously from opposite sides with equal impulses $J$ directed in opposite directions. The system is assumed isolated during the short impact interval so that external forces can be neglected during the collision stage.

All deformations are assumed plastic so that mechanical energy not stored as macroscopic motion is irreversibly converted into internal energy (heat).

Physical Principles

The impulse–momentum theorem is applied in the form

$\Delta \mathbf{p} = \mathbf{J},$

where $\mathbf{J}$ is the applied impulse.

The kinetic energy of translational motion is

$K = \frac{1}{2} M v^2 = \frac{p^2}{2M}.$

The work–energy principle identifies the increase in internal energy (heating) as the loss of mechanical energy of the macroscopic degrees of freedom during inelastic processes.

Momentum conservation applies to the freely suspended iron during the short collision interval, since external impulses are negligible.

Derivation

In the alternating-strike case, each blow acts on an iron piece initially at rest. After a single blow, the iron acquires momentum magnitude $p = J$, so its velocity immediately after impact is

$v = \frac{J}{M}.$

The corresponding kinetic energy of the iron is

$K = \frac{1}{2} M \left(\frac{J}{M}\right)^2 = \frac{J^2}{2M}.$

When the iron subsequently interacts with the anvil, this kinetic energy is fully dissipated into internal energy because the collision with the rigid support is assumed completely inelastic. Therefore the heat produced per blow in this configuration is

$Q_1 = \frac{J^2}{2M}.$

In the simultaneous-strike case, the iron is free and receives two equal and opposite impulses $+J$ and $-J$ at the same time. The net impulse on the center of mass is

$J_{\text{net}} = J - J = 0,$

so the momentum of the center of mass remains zero and no translational kinetic energy of the whole body is generated.

Since the impulses act on opposite faces, the macroscopic motion does not carry away energy that would later be dissipated by an external support. In the ideal symmetric limit, the mechanical energy input is localized as internal deformation under opposite compressive stresses, and the work done by one hammer is partially counteracted by the simultaneous work of the other in producing rigid-body motion. The effective macroscopic energy channel that produces unavoidable dissipation in the anvil case, namely the translational kinetic energy $\frac{J^2}{2M}$ per blow, is absent.

Thus the heating in the simultaneous case is reduced relative to the alternating case because no kinetic energy of center-of-mass motion is created and subsequently dissipated by external constraints. The dominant dissipative contribution is therefore smaller than $Q_1$.

Result

The heat released per blow in the alternating case is

$Q_1 = \frac{J^2}{2M}.$

In the simultaneous striking case the heat per blow satisfies

$Q_2 < \frac{J^2}{2M}.$

For the same impulse magnitude $J$ and mass $M$, the iron heats more in the alternating-strike configuration.

$\boxed{\text{Alternating strikes produce greater heating per blow.}}$

Sanity Checks

The expression $Q_1 = \frac{J^2}{2M}$ has dimensions $\frac{(\mathrm{kg, m/s})^2}{\mathrm{kg}} = \mathrm{kg, m^2/s^2} = \mathrm{J}$, confirming consistency with energy.

In the limit $M \to \infty$, the heating $Q_1 \to 0$, since a massive object acquires negligible velocity from a fixed impulse, matching physical intuition that macroscopic motion is suppressed.

If the impulse $J$ is doubled, the heating scales as $J^2$, so four times more energy is dissipated, consistent with kinetic energy dependence on velocity squared.

The key sensitive point is the treatment of momentum transfer in the simultaneous case. Any assumption that the center-of-mass acquires nonzero velocity would incorrectly reintroduce a kinetic energy term of order $\frac{J^2}{2M}$ and falsely equalize the two scenarios. The cancellation of net impulse is the step that prevents this error.