The simple Pendulum
A simple pendulum is an idealized model consisting of a point mass suspended by a massless, unstretchable string. When the point mass is pulled to one side of its straight-down equilibrium position and released, it oscillates about the equilibrium position.Familiar situations such as a wrecking ball on a crane’s cable or a person on a swing (Fig. 14.21a) can be modeled as simple pendulums.
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The path of the point mass (sometimes called a pendulum bob) is not a straight line but the arc of a circle with radius L equal to the length of the string (Fig. 14.21b). We use as our coordinate the distance x measured along the arc. If the motion is simple harmonic, the restoring force must be directly proportional to x or (because x = Lu) to u. Is it?
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Figure 14.21b shows the radial and tangential components of the forces on the mass. The restoring force FØ is the tangential component of the net force:
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Gravity provides the restoring force Fu; the tension T merely acts to make the point mass move in an arc. Since Fu is proportional to sin u, not to u, the motion is not simple harmonic. However, if angle u is small, sin u is very nearly equal to u in radians (Fig. 14.22). (When u = 0.1 rad, about 6°, sin Ø = 0.998. That’s only 0.2% different.) With this approximation, Eq. (14.30) becomes
The restoring force is then proportional to the coordinate for small displacements, and the force constant is k = mg/L. From Eq. (14.10) the angular frequency v of a simple pendulum with small amplitude is
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The corresponding frequency and period relationships are
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These expressions don’t involve the mass of the particle. That’s because the gravitational restoring force is proportional to m, so the mass appears on both sides of gF S = ma S and cancels out. (The same physics explains why bodies of different masses fall with the same acceleration in a vacuum.) For small oscillations, the period of a pendulum for a given value of g is determined entirely by its length.
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Equations (14.32) through (14.34) tell us that a long pendulum (large L) has a longer period than a shorter one. Increasing g increases the restoring force, causing the frequency to increase and the period to decrease.
The motion of a pendulum is only approximately simple harmonic. When the maximum angular displacement ? (amplitude) is not small, the departures from simple harmonic motion can be substantial. In general, the period T is given by
We can compute T to any desired degree of precision by taking enough terms in the series. You can confirm that when ? = 15°, the true period is longer than that given by the approximate Eq. (14.34) by less than 0.5%
A pendulum is a useful timekeeper because the period is very nearly independent of amplitude, provided that the amplitude is small. Thus, as a pendulum clock runs down and the amplitude of the swings decreases a little, the clock still keeps very nearly correct time.