Center of Mass, System Momentum, and Continuous Distributions

All flashcards

Discrete center of mass

For particles with masses $m_j$ at positions $\vec r_j$, $\vec r_{\mathrm{CM}}=\frac{1}{M}\sum_j m_j\vec r_j$, where $M=\sum_j m_j$.

System-level momentum law

The net external force equals the rate of change of total system momentum: $\vec F_{\mathrm{ext}}=\frac{d\vec p_{\mathrm{sys}}}{dt}$.

Continuous center of mass

For a continuous mass distribution, $\vec r_{\mathrm{CM}}=\frac{1}{M}\int \vec r\,dm$.

Internal force

A force arising from an interaction between parts included within the chosen system boundary.

Uniform thin hoop

Its center of mass is at its geometric center because opposite mass contributions cancel by symmetry.

Quiz questions

1. For a system of constant total mass, how is its center-of-mass acceleration related to the net external force?Answer: The net external force equals the total mass times the center-of-mass acceleration.
2. For particles with constant masses, which expression gives total system momentum in terms of center-of-mass velocity?Answer: $\vec p_{\mathrm{sys}}=M\vec v_{\mathrm{CM}}$
3. When calculating the center of mass, a particle lies at the coordinate origin. How should its mass be treated?Answer: Include it in the total mass, even though its position contribution is zero.
4. Where is the center of mass of a uniform thin hoop?Answer: At the hoop’s geometric center.
5. Must the center of mass of an object or system coincide with material belonging to it?Answer: No; it can lie in empty space or within material.