The Simple Pendulum

PENDULUMS

A simple pendulum is an idealized system in which a point mass is suspended at the end of a massless string. Figure 15.10 shows a simple pendulum length of L and a "pendulum bob" of mass m. The distance along the arc from the lowest point is s = Lθ, where θ is the angle (in radians) to the vertical. The net force on the bob along the tangent is the component of the weight along the tangent. Newton's second law applied along this direction is

The negative sign arises from the way s is defined. Its physical meaning is that the component of the weight acts as a restoring force. Clearly this equation does not have the form associated with SHM. However, for small angles sin θ ~θ, where θ is in radians. Since s = Lθ, we substitute and sin θ ~ θ into the above equation to find

(15.12)

Comparing this with Eq. 15.5 for SHM we see that within the small-angle approxi­mation, a simple pendulum executes simple harmonic motion with an angular frequency

FIGURE 15.10

(Simple pendulum)

and a period

(15.13)

The period does not depend on either the mass or the amplitude (Galileo's discov­ery). The solution to Eq. 15.12 is based on Eq. 15.2:

(15.14)

where θ₀ is the angular amplitude. Note that θ is a physical angular displacement, whereas ф is a mathematical phase constant that depends on the initial conditions. Furthermore, the angular frequency ω should not be confused with the (physical) instantaneous angular velocity

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