A pendulum 45cm long swings through a vertical angle of 30 degrees. Find the distance of the altitude through which the pendulum bob rises.

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quirozd | High School Teacher | (Level 3) Adjunct Educator

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Answer: 6.03 cm

One of the ways I like to answer these questions is by drawing state diagrams. A state diagram is just a "snap-shot" of the object in motion. In this case, you can overlap the two states to get a relationship to determine h, the altitude the pendulum bob goes.

Now that we have a clear picture, we will use trigonometry to solve this puzzle. The term vertical angle just means the angle swept from the vertical - a vertical angle of 30° corresponds to a 60° horizontal angle. 

When the pendulum is at its peak, it has a vertical component of `45text(cm)*cos(30)~~38.97text(cm)` . From the graphic above, that is the distance from the pivot to the green line.

When the bob is at its lowest position it is 45 cm away from the pivot. The altitude is just the difference of these two numbers.

`45text(cm)-45text(cm)*cos(30)~~6.03text(cm)`

BTW: We can look at this and determine the altitude, `h` , for any pendulum if we know the vertical angle, `theta` , and the length, `l` .

`h=l(1-costheta)`

Answer: 6.03 cm

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