Practice question · Multiple choice
Engineers replace sin θ with θ for small angles, and pendulum clocks work because of it. What is being assumed, and when does it break?
Hints
- Write the first two terms of the series for sin θ. What is the size of the one you dropped?
- Ask what happens to a pendulum's period at a 40° swing.
Show the answer
C. That the first Taylor term dominates for small θ, with error like θ³
Why
The discarded term is θ³/6, so at 0.1 radians the error is about one part in six hundred and the approximation is superb. Push to a wide swing and it fails, a pendulum's period does grow with amplitude, which is exactly why clock escapements are built to keep the arc small.
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