The Physics Philes, lesson 65: The Judges Give This Post a 9.8…Meters per Second per Second
Last week we started our discussion of equilibrium and elasticity with an quick overview of what equilibrium is. Today we’ll continue with a discussion of center of gravity.
Because we live in the real world and we want physics to be practical, we can’t ignore gravity. Which means that we need to take into account the weight of the body and the torque of that force. (Remember from way back when that weight is mass times gravity.) The weight of a body doesn’t act in one spot; it’s distributed. Hm. That’s a snag. Luckily, we can calculate the torque due to the body’s weight if we assume that the weight (ie, the entire force of gravity) is concentrated at a single point. We call this point the center of gravity.
We know that the force of gravity decreases as we go up in altitude, but if we can ignore this variation over the length of the body then the body’s center of gravity is the same as its center of mass.
Remember the definition of center of mass. it’s the result of the sum of all the particle’s masses and positions divided by the sum of the particle’s masses. The x, y, and z coordinates of the center of mass are the components of the position vector r. That means that the position vector of the center of mass is equal to the sum of the products of all the particle’s masses and position vector divided by the sum of the masses of the particles.
Clear as mud, right? I know, but stay with me.
Let’s assume that the acceleration due to gravity has the same magnitude and direction at every point in the body. Every particle in the body experiences a gravitational force and the total weight of the body is the vector sum of a large number of parallel forces. Any random particle in a body will have a weight of the particle’s mass times gravity. If the position vector of the particle with respect to any origin, then the torque vector of the weight with respect to the origin is the position vector times the product of the particle’s mass times gravity. That’s just for the one particle. But it holds true for all the particles. The total torque due to the gravitational force is the quantity of the total mass times the position vector multiplied by gravity.
Now, if we multiply and divide by the total mass, we can simplify and get a nice, neat, simple equation:
Hopefully you could follow my word picture to this point, because this is important. The total gravitational torque is the same as though the total weight were acting on the position vector of the center of mass, aka the center of gravity. If gravity has the same value at all points on the body, the center of gravity of the body is the same as the body’s center of mass.
With that bombshell, it’s time to end. I am, unfortunately, super pressed for time today. But don’t worry! Next week we’ll discuss how to actually use this new center of gravity knowledge you have.
Featured image credit: Wikimedia Commons