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Most of the time, 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 as 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 it 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 tangential 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. When you get a good looking graph, right-click on it to clone the graph and resize it for a better view.
Since in radians, x = θ L, 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. and for small enough angles we have simple harmonic motion.
Now consider both the motion of a pendulum and 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), and 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, why?) and play the animation. What do you notice about the graph? Do you see why the spring constant was carefully chosen?
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?
Illustration by Morten Brydensholt, Wolfgang Christian, and Mario Belloni
Script by Morten Brydensholt, Wolfgang Christian, and Mario Belloni
© 2003 by Prentice-Hall, Inc. A Pearson Company