# Dynamics of systems of material points

• I
• Hak
Hak
I have difficulty understanding the extension of the fundamental laws of material point dynamics to systems.

Example 1:
Consider a system consisting of two material points. Suppose that the two forces acting on the two constitute a pair of forces of nonzero arm. The resultant of the forces acting on the system is zero. The resultant moment is not! Why is it that to calculate the resultant moment we add up the moments of the two forces calculated separately, thus obtaining a different result than if we calculated the moment of the resultant of the forces (which would result in a null moment)?

Example 2:
Same system, but this time the pair of forces has zero arm. Assuming that due to the effect of the two (constant) forces, the two points move by a stretch ##s##, why is the total work of the two forces derived by summing the work of the individual forces, resulting in ##2Fs##, instead of calculating the work of the resultant of the forces (which would give work equal to ##0##)?

Hak said:
Why is it that to calculate the resultant moment we add up the moments of the two forces calculated separately, thus obtaining a different result than if we calculated the moment of the resultant of the forces (which would result in a null moment)?
Because we want the resultant moment to represent the rate of change of angular momentum, which can be non-zero, even if the net force is zero.

Hak said:
why is the total work of the two forces derived by summing the work of the individual forces, resulting in ##2Fs##, instead of calculating the work of the resultant of the forces (which would give work equal to ##0##)?
Because it takes a non-zero amount of energy to do this, so your approach would not make sense.

Hak said:
Consider a system consisting of two material points. Suppose that the two forces acting on the two constitute a pair of forces of nonzero arm. The resultant of the forces acting on the system is zero. The resultant moment is not! Why is it that to calculate the resultant moment we add up the moments of the two forces calculated separately, thus obtaining a different result than if we calculated the moment of the resultant of the forces (which would result in a null moment)?
A force by itself does not have a moment. Not even a zero one. In order to compute a moment you need the force and a reference point with respect to which to compute the moment. For a single particle it would be relatively natural to pick the particle itself as the reference point - thereby obtaining zero moment for any force with a line of action through that particle.

Now here is the thing: In order to add moments you must compute them relative to the same point, so once you picked a reference point you need to stick with it for all forces. Hence, in your two particle case, at least one of the forces will provide non-zero moment.

Note: The total moment generally depends on the point of reference. However, this is not the case if the net force is zero as in your case.

Hak said:
Same system, but this time the pair of forces has zero arm. Assuming that due to the effect of the two (constant) forces, the two points move by a stretch s, why is the total work of the two forces derived by summing the work of the individual forces, resulting in 2Fs, instead of calculating the work of the resultant of the forces (which would give work equal to 0)?
Because the work done by a force is the force multiplied by the displacement of what it is acting on - the particles in this case. The forces are opposite but so are the displacements of the particles. Hence 2Fs.

vanhees71

## What are the dynamics of systems of material points?

The dynamics of systems of material points refer to the study of how groups of particles or points, which have mass but negligible size, move and interact under the influence of forces. This field involves analyzing how these forces affect the motion of each point and the system as a whole, often using principles from Newtonian mechanics, conservation laws, and differential equations.

## How do you describe the motion of a system of material points?

The motion of a system of material points is described by the positions, velocities, and accelerations of each point in the system. These quantities are typically functions of time and can be determined using Newton's laws of motion. Additionally, the interactions between the points, such as gravitational, electromagnetic, or contact forces, play a crucial role in defining the overall motion.

## What is the significance of the center of mass in a system of material points?

The center of mass is a critical concept in the dynamics of systems of material points. It represents the average position of all the mass in the system, weighted by their respective masses. The motion of the center of mass can simplify the analysis of the system's dynamics because it behaves as if all the external forces act on it, and it follows a trajectory governed by the net external force acting on the system.

## How are conservation laws applied to systems of material points?

Conservation laws, such as the conservation of momentum, energy, and angular momentum, are fundamental principles in the dynamics of systems of material points. These laws state that in the absence of external forces or torques, the total momentum, energy, and angular momentum of the system remain constant. These principles are used to solve problems and predict the future behavior of the system by providing constraints and relationships between the physical quantities involved.

## What role do differential equations play in the dynamics of systems of material points?

Differential equations are essential tools in the dynamics of systems of material points. They describe how the positions, velocities, and accelerations of the points change over time under the influence of forces. By solving these differential equations, often derived from Newton's second law of motion, scientists can determine the trajectories and behavior of the system. These equations can be solved analytically for simple systems or numerically for more complex interactions.

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