Illustration 8.8: Moving Objects and Center of Mass
Please wait for the animation to completely load.
A green block, 1.00 kg, sits on a red block, 4.46 kg, as shown in the animation (position is given in meters and time is given in seconds). Restart. All surfaces are frictionless except for the gray patches on the red block. Given the self-propelled motion of the green block in Animation 1, are momentum and energy conserved in the animation? If not, why not?
Well, momentum is conserved because there are no external forces. The momentum of the system was zero before the green block moved, is zero when the blocks move, and is again zero when the blocks are stationary. From the point of view of the center of mass, Vcm = 0 m/s and therefore Pcm = 0 kg·m/s.
We can see this by considering what happens to the center of mass during Animation 2. The center of mass of the system is Xcm = (m1x1 + m2x2)/(m1 + m2) and is represented by the black dot. Note that the center of mass of the system does not move relative to the ground but does move to the left relative to the right edge of the red block as the red block moves to the right. In fact we can look at the center of mass for each object by replacing each block by a dot as well as shown in Animation 3.
What about energy? As is always the case, whether energy is conserved depends on how you define your system. Looking just at the center of mass, since Vcm = 0 m/s, energy is conserved. However, if we look at the blocks individually, energy (in the sense of mechanical energy) is not conserved. Energy stored in the individual elements of the system (presumably the green block's potential energy) is turned into kinetic energy of both blocks and then is dissipated by friction.