The Physics Philes, lesson 83: Dinosaur Walking
Physical phenomena, as I’ve learned from just my brief venture into its depths, are complicated. Luckily, we have ways of kind of easing our way into understanding these things. Physicists can answer simple questions using “idealized models.” Once those easy questions are answered, we can progressively complicate the model to gain a deeper understanding.
So far in our discussion of oscillation, we’ve been using idealized models. Last week’s simple pendulum used an idealized model. This week we complicate the picture a bit with what physicists call the physical pendulum.
While the simple pendulum has all the mass concentrated a single point, a physical pendulum uses an extended body. What’s kind of cool about the physical pendulum is that, even though you might think it would be a much more complicated problem, it’s actually not much more difficult than the simple pendulum. As least for small oscillations. Here’s a drawing that kind of looks like a big green booger but is actually supposed to be an irregularly shaped physical pendulum:
O represents the axis around which the body rotates, or the pivot point. The center of gravity is represented by cg. Gravity acts on the body there. When the body is in its equilibrium position, the center of gravity is directly below the pivot point O. However, the body is displaced by the angle θ. The distance d is the distance from the pivot point to the center of gravity and the total mass is denoted by m. The blue dotted line is the trajectory of the center of gravity as it oscillates about the pivot point.
When a body is displaced like this, it’s own weight causes a restoring torque, which can be expressed as
Remember that mg is the body’s weight. It’s negative so we know that the torque is in the opposite direction of the displacement.
Notice that, like the simple pendulum, the oscillation isn’t simple harmonic because the torque is proportional to sinθ instead of just θ. But, again like the simple pendulum, if θ is small enough, we can approximate sinθ by θ in radians. So the oscillation is approximate simple harmonic, and we can use this equation:
We can also express this in terms of the moment of inertia. Remember that the equation for the moment of inertia is
The sum of all torques equals the moment of inertia times the acceleration. That means that we can equate the moment of inertia equation to the torque of our pendulum. Since acceleration is equal to rate of change of velocity over time, our oscillation equation is also equal to the moment of inertia times the second derivative of θ with respect to time.
If we manipulate this a bit, we see something interesting:
When we first started talking about simple harmonic motion, we found that the acceleration of a spring mass system in simple harmonic motion is
As you can see, (k/m) in this equation plays the same role as (mgd/I) plays. This means that we can express the angular frequency as:
The frequency is 1/2π times the angular frequency, so the period is:
This period equation is kind of important because it’s used as the basis of a common method for determining the moment of inertia for a body with a complicated shape experimentally. To do this, you need to do four things. First, you need to find the center of gravity of the oddly-shaped object by balancing it. Second, you need to suspend the body so it’s free to oscillate about the axis. Third, you need to measure the period. And fourth, you just use the period equation above to calculate the moment of inertia.
This sounds esoteric, but it’s actually pretty useful in fields outside of physics. For example, knowing the moment of inertia is important for biomechanics researchers to analyze how animals like dinosaurs walked. Walking animals have a natural walking pace. If we consider the natural walking speed to correspond to the oscillation of a physical pendulum. Pretty cool!
We’re getting towards the end of our discussion of oscillations. Next week we’ll start in on damped oscillations.
Featured image credit: John Martinez Pavliga via Flickr