Stopped Clock...

This morning I noticed that the clock in our laundry room (at right) had stopped.  The second hand was wiggling once per second as the clock tried to advance, but failed.  Clearly the battery needs to be changed.

But this brought to mind a question I asked myself some years ago, and figured out: Why do most battery-powered clocks seem to stop at a particular time (generally either around  8:45:45 or 7:45:45)?

The answer lies in the unbalanced hands, especially the second hand and the minute hand.  By “unbalanced” I mean that if those hands were free to rotate they'd rotate so that the heaviest portion was pointing straight down.  Some clocks are made with balanced hands (this was particularly common on old-fashioned wind-up or weight-driven clocks) – either by making the short side of the hand wider, or by putting added weight on the short side.

When the hand is unbalanced, as in our laundry room clock, the tiny electric motor in the clock has to work harder while the hands are on the left side of the clock than it does when they're on the right – it's fighting gravity in one case, and gravity is helping in the other.  This effect is most pronounced for the second hand, which takes about 60 times as much force to move as does the minute hand.  The hour hand takes about 60 times less than the minute hand, or 3600 times less than the second hand.

The force required to move the second hand is at it's absolute highest when it's pointing directly to the left.  When it's pointing in this direction, the force of gravity has the most leverage with respect to the shaft that the motor must turn.  So...as the battery gradually discharges, the motor can produce less and less force.  As the second hand sweeps around the clock from the 12 o'clock position, at first little force is required (because gravity is helping).  As it passes the 6 o'clock position, more and more force is required (because the motor is lifting the hand and gravity is making it harder).  So as the battery discharges, the force required to move the hand is most likely to exceed what the motor can provide when the second hand moves to the 9 o'clock position.

Mathematically, the additional force required to move an unbalanced clock hand imposed by gravity can be described as:

-k sin(θ)

where k is determined by the degree to which the hand is out-of-balance, and θ is the angle of the hand (where 0 is the 12 o'clock position).  At 45 seconds after the minute, the second hand is at 270°.  One second before that, it's at 264°.  The sines for those angles are -1 and -0.99452, so there's about a half-percent more force required to move the second hand between the 45th and 46th second than between the 44th and 45th – and that little bit of extra force is why cheap battery-powered clocks stop at around the same time...

On thinking about this from an engineering perspective, it occurred to me that there are two very simple ways to extend the battery life of these clocks:

1. Balance the hands.
2. Remove the second hand.

Of course, the batteries last for years any way, so it probably makes no practical difference...