2.1 Systems and Center of Mass

A system is whatever you draw an imaginary boundary around: one block, two blocks tied together, or a ball plus Earth. Everything outside is the environment.

The choice decides which forces are internal (between parts of the system) and which are external (from the environment). Only external forces change the motion of the system as a whole. For two blocks joined by a string, choosing both blocks as the system makes the tension internal, so it drops out. Choosing one block alone makes the tension external, which is how you then find it.

If the system's internal structure doesn't matter to the question, you can model it as a single object, a point with all the mass, located at the center of mass.

The center of mass is a weighted average of position, where heavier parts count more:

$$x_{cm} = \frac{\sum m_ix_i}{\sum m_i}$$

Do the same for y (and z) separately. The center of mass of two objects lies on the line between them, closer to the heavier one, and it splits the distance in the inverse ratio of the masses.

For a symmetric object with mass spread evenly (uniform), the center of mass lies on every line of symmetry. A uniform rod's is at its midpoint, and a uniform disk's or sphere's is at its center. The center of mass doesn't have to be inside the material: a ring's is in the empty middle.

For a continuous object, chop it into tiny pieces of mass dm and add them with an integral:

$$x_{cm} = \frac{\int x\,dm}{\int dm}$$

The bottom integral is just the total mass M. To integrate over position, you need to write dm in terms of dx. For a thin rod along the x-axis, the linear mass density $\lambda = \frac{dm}{dx}$ is the mass per unit length (in kg/m), so $dm = \lambda\,dx$. Then:

$$M = \int_0^L \lambda\,dx, \qquad x_{cm} = \frac{1}{M}\int_0^L x\lambda\,dx$$

If λ is constant (a uniform rod), this gives L/2, as symmetry says it should. If λ grows along the rod, like λ = bx, the center of mass shifts toward the denser end. Keep λ inside the integral whenever it depends on x.

The center of mass is the point that moves as if all the mass were there and all the external forces acted on it. A wrench spinning as it slides across ice wobbles in a complicated way, but its center of mass moves in a straight line at constant speed. This idea comes back in Newton's second law (2.5) and in momentum (4.3).