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Pendulums
by AudioLearn
PENDULUMS
Figure 98.
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A simple pendulum is defined as an object swinging or suspended from a light wire or string. Pendulums have the ability to perform simple harmonic motion. Displacement in a pendulum is related to the arc length, defined by the letter s. The net force on the object will be the tangent to the arc, which will be equal to mass times the force of gravity times the sine of theta. The tension on the string will exactly cancel the component mass times the force of gravity times the cosine of theta, which is parallel to the string at all times. Figure 99 shows the pendulum swinging:
Figure 99.
As long as the restoring force is directly proportional to the displacement, this would be a simple harmonic oscillator. For small angles less than 15 degrees, the sine of theta will be roughly equal to theta, which means that the restoring force will be mass times the force of gravity times theta. Theta will be expressed in radians so that the displacement will equal the length of the cord on a pendulum times the angle theta in radians. This means is that the force will be roughly equal to the mass times the force of gravity times the length of the cord times the arc length. These equations are seen in figure 100:
Figure 100.
What figure 100 ties into is the force constant k, which is mass times the force of gravity divided by the length. For angles less than 15 degrees, this will behave according to Hooke’s law and the restoring force will be directly proportional to the displacement. This will make the pendulum a simple harmonic oscillator when the angle of displacement is less than 15 degrees.
As for the energy of a simple harmonic oscillator, the energy will be shared between the elastic potential energy and the kinetic energy. What we already know is that this will equal one-half multiplied by the mass times the square of the velocity (kinetic energy) and one-half times the force constant times the displacement squared (the potential energy). The sum of these two things will be constant and will be shown in figure 101:
