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Illustration 16.2: The Simple Pendulum and Spring Motion
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When we think about simple harmonic motion we think about a mass on a spring. This is the prototype motion and is the easiest to deal with since k, the spring constant, is the proportionality factor between F and -x. However, there is another standard example of simple harmonic motion that is all around us, that of pendulum motion. Restart. A pendulum is nothing more than a heavy object (the pendulum bob) hanging from a very light string (if the string's mass is large enough, we have a compound pendulum and the string must be considered). Consider Animation 1. Here the length of the string is 15 m and the mass of the pendulum bob is 1 kg (position is given in meters, angle is given in radians, and time is given in seconds). When we analyze the forces acting on the pendulum bob (drag the pendulum bob from its equilibrium position and press "play"), we find that the force of gravity and the force of tension act. The simplest way to analyze these forces is to consider their effect in the radial direction and the direction tangent to the circular path of the pendulum. The part of the gravitational force opposite to the tension must cancel the force of the tension in the string when the pendulum is at rest. However, when the pendulum bob is moving, the tension must be greater to provide the centripetal force required. This leaves the component of the force of gravity perpendicular to the tension and tangent to the path of the pendulum. Show Animation 1 with pendulum bob path. When we do the calculation, we find that the net force on the pendulum bob goes like
Ftan = - mg sin(θ),
which at first glance does not look at all like simple harmonic motion. But what happens when the angle θ is small? Well, sin(θ) ≈ θ for small enough θ; therefore, Ftan small angles = - mg θ.
Drag the pendulum bob to a large angle and see how the two tangential forces (any angle vs. small angle) deviate at large angles. The motion of the pendulum is shown according to the actual force, Ftan = - mg sin(θ), and not the small angle approximation, Fnet = - mg θ, although both are shown on the graph. Therefore the period of the pendulum is the actual period. When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.
Since we are using radians, x = θ L, and the tangential force for small angles can be written as Ftan small angles = - (mg /L) x, where the proportionality factor between F and -x is now mg /L. For small enough angles (when sin(θ) ≈ θ) we have simple harmonic motion.
Now consider both the motion of a pendulum and the motion of a mass attached to a spring by looking at Animation 2. In this animation the pendulum is the same as Animation 1 (the net force on the bob is shown as a green arrow), the spring has a spring constant of 1.30666 N/m, and the mass of the red ball attached to the spring is 2 kg (the net force on the red ball is represented by the blue arrow). It may seem strange that we have chosen such an oddly precise value for the spring constant. Drag the pendulum to about 0.15 radians and drag the mass on the spring to some initial amplitude (it does not matter what this value is, but for simplicity chose 2.3 m) and play the animation. What do you notice about the graph? Do you see why the spring constant was carefully chosen? These values were chosen to tune the motion of the two systems to be the same:
ωmass-spring = (k/m)0.5 = ωpendulum = (keffective/m)0.5 = (g/L)0.5.
Now reset this animation and drag the pendulum bob to 0.75 radians and the mass on the spring to 10.3 m and play the animation. What happens now? By looking at Animation 1 can you say why this is? Notice as time goes on, that the two motions now deviate from each other. Large-amplitude pendulum motion is no longer simple harmonic motion.
Illustration authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.
Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.