Old Watch

Lenni has an old mechanical watch in pristine condition that has an internal balance wheel that operates perfectly. She takes a drive into the mountains. Will the watch run fast or slow?

Answer

The old watch will run fast. The balance wheel is the basic component that oscillates “exactly” 300 times for each minute on the watch face that is, the wheel changes direction 10 times each second! The moment of inertia of the balance wheel depends on how much ambient air is dragged along during each oscillation. The source of energy is a wound spring that is essentially unaffected by the air because of its extremely small change in configuration. In the mountains, the viscosity and density of the air decrease slightly, allowing the balance wheel to oscillate faster. Newton’s second law applied to this rotational motion is required.

From its momentary stop to change direction, the balance wheel must accelerate to its maximum angular velocity, then accelerate back to rest, etc. The net torque τ equals the moment of inertia I times the angular acceleration α—that is, τ = Iα. The moment of inertia is determined by the mass distribution with respect to the rotation axis, and the air carried along with the balance wheel motion adds to the moment of inertia of the wheel alone. That is why the watch must be recalibrated when the location of the owner is at a different elevation than the factory.

When the balance wheel drags along less air mass at a higher elevation, the moment of inertia is less for the same net torque, so the angular acceleration is greater. Less time is needed to reach top angular speed, and less time is needed to come to rest again.

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