The Physics Philes, lesson 38: Outside Forces
In which momentum is not conserved, external forces are exerted, and parabolic trajectories are maintained.
Are you ready to finally finish up the basics of momentum and center of mass? Well hold on to your hats, because it’s time to roll.
Last week we discussed momentum and center of mass in systems where no external forces acted on the body. But what if the net external fore on the system is not zero? In that case, total momentum is not conserved and the velocity of the center of mass changes.
When momentum is conserved, we’ve seen how the velocity of the center of mass can be expressed in terms of the velocities of the individual particles. Well guess what! Acceleration works in the same way. The product of the center of mass and the acceleration of the center of mass is the sum of the products of the acceleration and masses of the individual particles. This is also equal to the vector sum of the net forces for all the forces on all the particles – external and internal. But because of Newton’s third law, those internal forces cancel each other out, so we’re left with only the external forces. Thus, the net external force is equal to the product of the center of mass and the acceleration of the center of mass.
The rule, then, is this:
When a body or a collection of particles is acted on by external forces, the center of mass moves just as though all the mass were concentrated at that point and it were acted on by a net force equal to the sum of the external forces on the system.
This rule is incredibly important. Do you remember, a long time ago, when we were learning how to draw helpful diagrams? The only reason why we can solve Newton’s law problems by assuming, say, a sled, is a single point sitting on a plane of frictionless ice is because of this rule.
There’s one more property of center of mass I should probably point out before we leave the subject, and that’s it’s propensity to continue along a parabolic trajectory. Let’s think about a cannon shell that’s been launched into the air. It’s traveling on a parabolic trajectory and explodes mid-flight. The two pieces will go off on a new trajectory, but the trajectory of the center of mass will stay the same; it will continue to travel along the original path.
This is important when we have to look at the motion of rigid bodies, and it comes in handy in astronomy. For example, did you know that it’s not correct to say that the moon orbits the sun. The moon and the sun actually orbit around their center of mass. (Is your mind blown right now, too?)
On that bombshell, it’s time to go. Next week we’ll start a discussion of the rotation of rigid bodies and angular velocity. See you then!
Featured image credit: Joe Shlabotnik