15.6.3 The Physical Pendulum

A mass on a string is often called a simple pendulum. But you can also make a pendulum from any solid object that swings back and forth on a pivot under the influence of gravity. This is called a physical pendulum.

FIGURE 15.20 shows a physical pendulum of mass M for which the distance between the pivot and the center of mass is l. The moment arm of the gravitational force acting at the center of mass is $d = l sin θ$ , so the gravitational torque is

τ = - M g d = - M g l sin θ

The torque is negative because, for positive $θ$ , it’s causing a clockwise rotation. If we restrict the angle to being small $(θ < 10 °)$ , as we did for the simple pendulum, we can use the small-angle approximation to write

τ = - M g l θ

(15.49)

Gravity exerts a linear restoring torque on the pendulum—that is, the torque is directly proportional to the angular displacement $θ$ —so we expect the physical pendulum to undergo SHM.

The figure illustrates a physical pendulum depicted as a stick with an encircled x-mark at its center.

FIGURE 15.20 Full Alternative Text

From Chapter 12, Newton’s second law for rotational motion is

α = \frac{d^{2} θ}{d t^{2}} = \frac{τ}{I}

where I is the object’s moment of inertia about the pivot point. Using Equation 15.49 for the torque, we find

\frac{d^{2} θ}{d t^{2}} = \frac{- M g l}{I} θ

(15.50)

The equation of motion is of the form $d^{2} θ / d t^{2} = - C θ$ , so the model for simple harmonic motion tells us that the motion is SHM with angular frequency

ω = 2 π f = \sqrt{\frac{M g l}{I}}

(15.51)

It appears that the frequency depends on the mass of the pendulum, but recall that the moment of inertia is directly proportional to M. Thus M cancels and the frequency of a physical pendulum, like that of a simple pendulum, is independent of mass.

EXAMPLE 15.9 A swinging leg as a pendulum

A student in a biomechanics lab measures the length of his leg, from hip to heel, to be 0.90 m. What is the frequency of the pendulum motion of the student’s leg? What is the period?

MODEL We can model a human leg reasonably well as a rod of uniform cross section, pivoted at one end (the hip) to form a physical pendulum. For small-angle oscillations it will undergo SHM. The center of mass of a uniform leg is at the midpoint, so $l = L / 2$ .

SOLVE The moment of inertia of a rod pivoted about one end is $I = \frac{1}{3} M L^{2}$ , so the pendulum frequency is

f = \frac{1}{2 π} \sqrt{\frac{M g l}{I}} = \frac{1}{2 π} \sqrt{\frac{M g (L / 2)}{M L^{2} / 3}} = \frac{1}{2 π} \sqrt{\frac{3 g}{2 L}} = 0.64 Hz

The corresponding period is $T = 1 / f = 1.6 s$ . Notice that we didn’t need to know the mass.

REVIEW As you walk, your legs do swing as physical pendulums as you bring them forward. The frequency is fixed by the length of your legs and their distribution of mass; it doesn’t depend on amplitude. Consequently, you don’t increase your walking speed by taking more rapid steps—changing the frequency is difficult. You simply take longer strides, changing the amplitude but not the frequency.