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# CES Mechanics II Lecture 10

RWTH Aachen University, Bachelor program in Computational Engineering Science

Topics:

• Particle systems 2.1

The system of n particles with masses m_i and positions \mathbf{r}_i is characterized by the center of mass (CM, Massenmittelpunkt) and center of gravity (CG, Schwerpunkt); the two points are the same in a constant gravitational field and are located at:

\bar{\mathbf{r}} = \sum_{i=1}^n m_i \mathbf{r}_i / \sum_{i=1}^n m_i.

This can be understood as weighted average of particle locations with the weights corresponding to the masses. Another form of this definition is:

m \; \bar{\mathbf{r}} = \sum_{i=1}^n m_i \mathbf{r}_i,

where m = \sum_{i=1}^n m_i is the total mass of the system.

The scalar version is:

m \; \bar{x} = \sum_{i=1}^n m_i x_i,

m \; \bar{y} = \sum_{i=1}^n m_i y_i,

m \; \bar{z} = \sum_{i=1}^n m_i z_i.

Differentiating with respect to time one obtains the velocity and acceleration of the CM:

m \; \bar{\mathbf{v}} = \sum_{i=1}^n m_i \mathbf{v}_i,

m \; \bar{\mathbf{a}} = \sum_{i=1}^n m_i \mathbf{a}_i.

• Kinetics of individual particles

Just like for a single particle, equations of motion can be written for each particle in the system using Newton's second law. We will distinguish here between external forces \mathbf{F}_i coming from outside the system and internal forces \mathbf{f}_{ij} acting between particles. The internal forces only occur in pairs of equal and opposite vectors \mathbf{f}_{ij} = - \mathbf{f}_{ji} because of action is equal to reaction. A particle cannot exert a force on itself, so \mathbf{f}_{ii} = \mathbf{0}. Consequently we can write n equations:

\mathbf{F}_i + \sum_{j=1}^n \mathbf{f}_{ij} = m_i \mathbf{a}_i.

• Kinetics of the center of mass

Adding the above equations together, and recognizing that the internal forces cancel each other in the resulting sum, we can write the equation for the motion of the center of mass:

\sum_{i=1}^n \mathbf{F}_i = m \; \bar{\mathbf{a}}.

Note that only external forces influence the motion of the center of mass. This approach is particularly useful if the external forces are simple, internal forces unknown and motion of the CM uncomplicated. The alternative approach of kinetics of individual particles is useful when the motion of those particles is simple (e.g., rectilinear).

Corresponding Gross et al. Kinetik chapters are shown in blue.

Key techniques:

• finding center of mass of a particle system
• analyzing motion of the center of mass using external forces
• analyzing motion of the individual particles under external and internal forces

ConcepTests (RWTH only):

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