Let the total energy of a system of N particles be measured...

Let the total energy of a system of $N$ particles be measured in an arbitrary frame of reference, such that $K=\Sigma \frac{1}{2} m_{n} v_{n}^{2} .$ In the center-of-mass reference frame, the velocities are $v_{n}^{\prime}=v_{n}-v_{\mathrm{cm}}$, where $v_{\mathrm{cm}}$ is the velocity of the center of mass relative to the original frame of reference. Keeping in mind that $v_{n}^{2}=\overrightarrow{\mathbf{v}}_{n} \cdot \overrightarrow{\mathbf{v}}_{n}$, show that the kinetic energy can be written $$K=K_{\mathrm{int}}+K_{\mathrm{cm}}$$ where $K_{\text {int }}=\Sigma \frac{1}{2} m_{n} v_{n}^{\prime 2}$ and $K_{\mathrm{cm}}=\frac{1}{2} M v_{\mathrm{cm}}^{2} .$ This demonstrates that the kinetic energy of a system of particles can be divided into an internal term and a center-of-mass term. The internal kinetic energy is measured in a frame of reference in which the center of mass is at rest; for example, the random motions of the molecules of gas in a container at rest are responsible for its internal translational kinetic energy.

Let the total energy of a system of $N$ particles be measured in an arbitrary frame of reference, such that $K=\Sigma \frac{1}{2} m_{n} v_{n}^{2} .$ In the center-of-mass reference frame, the velocities are $v_{n}^{\prime}=v_{n}-v_{\mathrm{cm}}$, where $v_{\mathrm{cm}}$ is the velocity of the center of mass relative to the original frame of reference. Keeping in mind that $v_{n}^{2}=\overrightarrow{\mathbf{v}}_{n} \cdot \overrightarrow{\mathbf{v}}_{n}$, show that the kinetic energy can be written
$$K=K_{\mathrm{int}}+K_{\mathrm{cm}}$$
where $K_{\text {int }}=\Sigma \frac{1}{2} m_{n} v_{n}^{\prime 2}$ and $K_{\mathrm{cm}}=\frac{1}{2} M v_{\mathrm{cm}}^{2} .$ This demonstrates
that the kinetic energy of a system of particles can be divided into an internal term and a center-of-mass term. The internal kinetic energy is measured in a frame of reference in which the center of mass is at rest; for example, the random motions of the molecules of gas in a container at rest are responsible for its internal translational kinetic energy.

Key Concepts

Decomposition of Kinetic Energy

This concept involves splitting the total kinetic energy of a system into two distinct parts: one associated with the motion of the center of mass and the other with the internal motions of the particles relative to the center of mass. It enables a clearer analysis of how energy is distributed in a multi-particle system, particularly when distinguishing between overall translational motion and random internal motions.

Center-of-Mass Reference Frame

The center-of-mass frame is a special inertial frame where the total momentum of the system is zero, meaning the center of mass is stationary. Measuring particle velocities relative to this frame simplifies calculations, as it isolates the internal dynamics of the system from its collective movement.

Internal Kinetic Energy

Internal kinetic energy refers to the kinetic energy arising from the individual motions of particles relative to the center of mass. It accounts for the random or relative motions within the system and is independent of the overall translational motion of the system as a whole.

Relative Velocity Transformation

This concept involves expressing the velocity of each particle as the sum of the center-of-mass velocity and the particle’s velocity relative to the center of mass. Utilizing properties of vector algebra, specifically the dot product, allows for the expansion and separation of energy contributions from the internal and center-of-mass motions.

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