On a planet with gravitational field strength 3.5 m/s ^ 2, our intrepid space explorer observes an aardvark with unknown intentions idly swinging a rock back and forth at the end of a string.
Our explorer can't remember the formula that relates the period of a pendulum to the gravitational acceleration and the pendulum length. However, she had a physics course in which she learned to think, so she reasons it out as follows:
The force required to restore the pendulum is its weight multiplied by the ratio between its distance from equilibrium and the length of the pendulum. This seems right, because it ensures that the force increases with distance from equilibrium. It also seems right because the horizontal component of string tension should be in this proportion to the vertical tension (for small swings).
Thus F = (mg)(x/L), where x is the distance from equilibrium. The force constant is therefore
k = F/x = mg/L.
Knowing that a key to understanding simple harmonic motion is the fact that angular frequency = `sqrt(k/m), she quickly determines that the angular frequency is
`omega = `sqrt[(mg/L)/(m)] = `sqrt(g/L).
She knows that the angular frequency is the number of radians per second, and of course she has known for a very long time that there are 2 `pi radians in a complete cycle. So she reasons out that the angular frequency of the pendulum is 2 `pi rad / ( 3.4 sec) = 1.848 radians/second.
Setting this equal to `sqrt(g/L) and solving for L she realizes that
L = ( 3.5 m/s ^ 2) / ( 1.848 radians/second) ^ 2 = 1.024859 meters,
or roughly .6832394 times her height. This gives her some information with which to assess the advisability of an encounter with the pendulum's operator.
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