The Cycloid Scott Morrison
The cycloid Scott Morrison “The time has come”, the old man said, “to talk of many things: Of tangents, cusps and evolutes, of curves and rolling rings, and why the cycloid’s tautochrone, and pendulums on strings.” October 1997 1 Everyone is well aware of the fact that pendulums are used to keep time in old clocks, and most would be aware that this is because even as the pendu- lum loses energy, and winds down, it still keeps time fairly well. It should be clear from the outset that a pendulum is basically an object moving back and forth tracing out a circle; hence, we can ignore the string or shaft, or whatever, that supports the bob, and only consider the circular motion of the bob, driven by gravity. It’s important to notice now that the angle the tangent to the circle makes with the horizontal is the same as the angle the line from the bob to the centre makes with the vertical. The force on the bob at any moment is propor- tional to the sine of the angle at which the bob is currently moving. The net force is also directed perpendicular to the string, that is, in the instantaneous direction of motion. Because this force only changes the angle of the bob, and not the radius of the movement (a pendulum bob is always the same distance from its fixed point), we can write: θθ&& ∝sin Now, if θ is always small, which means the pendulum isn’t moving much, then sinθθ≈. This is very useful, as it lets us claim: θθ&& ∝ which tells us we have simple harmonic motion going on.
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