15.6 The Pendulum

Now let’s look at another very common oscillator: a pendulum. FIGURE 15.17a shows a mass m attached to a string of length L and free to swing back and forth. The pendulum’s position can be described by the arc of length s, which is zero when the pendulum hangs straight down. Because angles are measured ccw, s and $θ$ are positive when the pendulum is to the right of center, negative when it is to the left.

The figure labeled ("a") illustrates an oscillating spring.

The figure labeled (b) illustrates forces acting on an oscillating spring.

Full Alternative Text

Two forces are acting on the mass: the string tension $\vec{T}$ and gravity $\vec{F}_{G}$ . As we did with circular motion, it will be useful to divide the forces into tangential components, parallel to the motion, and radial components parallel to the string. These are shown on the free-body diagram of FIGURE 15.17b.

Newton’s second law for the tangential component, parallel to the motion, is

(F_{net})_{t} = \sum F_{t} = (F_{G})_{t} = - m g sin θ = m a_{t}

(15.44)

Using $a_{t} = d^{2} s / d t^{2}$ for acceleration “around” the circle, and noting that the mass cancels, we can write Equation 15.44 as

\frac{d^{2} s}{d t^{2}} = - g sin θ

(15.45)

This is the equation of motion for an oscillating pendulum. The sine function makes this equation more complicated than the equation of motion for an oscillating spring.