# Does air resistance have a greater affect on longer pendulums than shorter ones due to the greater exposure it has to air?

I'm thinking, because longer pendulums travel for a greater time period through the medium of air, air resistance will have a greater affect on the pendulum bob (and string) and so the angle the pendulum travels back at would be a lot less compared to that of a smaller length pendulum?

I have uploaded a picture to help explain my question.

In case my photograph is blurry - In the drawing a pendulum is released. Then the angle that it travels to on the opposite side is recorded, so is the distance from the centre of mass of the pendulum to the hypothetical trajectory (if pendulum were to swing in a complete vacuum) distance labelled "X". If length of pendulum were to be increased, would X remain constant? Or would the angle remain constant? Or perhaps, because of air resistance, would the angle decrease proportionally to the increasing of length of the pendulum?

Air drag typically increases approximately like $$v^2$$. The velocity of the pendulum scales like $$L^{1/2}$$. Therefore the force probably scales like $$L$$, and the work per oscillation like $$L^2$$. The energy in the pendulum is proportional to $$L$$, so the fraction of the energy dissipated per cycle should scale like $$L^2/L=L$$. The decrease in angle per cycle should therefore also increase in proportion to $$L$$.

• Could the downvoter explain their reason? If my answer is wrong, I'd like to learn from my mistake.
– user4552
May 27, 2019 at 14:49

If the experiment was carried out in a vacuum then air resistance would not play a part. For sufficiently small angles, the period of the pendulum is independent of the amplitude of swing. If the length was increased, the period would also increase, but this would not affect the amplitude.

Therefore, if the release angle is constant, the amplitude is constant.

• Thanks for such a quick reply. so if the release angle were to be 30 degrees from vertical and then swing to the other side 29 degrees from vertical. This would remain constant regardless of how long I make the pendulums length, if mass were to remain the same? May 26, 2019 at 9:59
• Yes, though the angle would remain at 30 degrees and would not decrease. May 26, 2019 at 10:12
• The first and second sentences are irrelevant. The third sentence just seems to state that amplitude and length can be controlled independently, which is obvious regardless of whether the s.h.m. approximation holds. The final sentence is a non sequitur.
– user4552
May 26, 2019 at 18:03