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.

Transcript

00:01 We are given a system of n particles.

00:08 Okay? and in this system of n particles, we have two reference frames.

00:20 One is the center of mass reference frame, abbreviating center of mass as com and frame of reference as fom.

00:28 Okay? and another is some arbitrary reference frame or arbitrary frame of reference.

00:36 Okay? now, for each of these particles in the center of mass reference frames, their velocity for the first particle, if we denote it by v1 dash, for the second particle, let it be v2 dash and proceeding so on till the nth particle whose velocity it will be vn dash.

00:58 Okay? and in the arbitrary frame of reference for these n particles only, let the velocity of the first particle be v1, the second particle be v2.

01:08 Similarly, carrying on, we reach the nth particle whose velocity is vn.

01:14 Okay? now, we define so and in this arbitrary frame of reference, let the velocity of the center of mass, okay, of the system be equal to where this vcm, sorry, v of cm denotes velocity of center of mass in this frame.

01:48 By this frame, i mean the arbitrary frame of reference.

01:53 From this, we can conclude that we are talking about the arbitrary frame of reference.

01:58 Okay? now, since if we were to express the velocity of the ith particle in the center of mass reference frame, then from these two definitions, it would be nothing but the sum of the velocity of the ith particle in the arbitrary frame of reference, okay, minus the velocity of the center of mass.

02:26 Okay? now, we define two terms.

02:34 The first term is for the kinetic, internal kinetic energy in the center of mass reference frame k of i and t.

02:43 Okay? this will be equal to half of mi, the, okay, total kinetic energy.

02:53 So, it will be the summation of, summation from the first particle to the nth particle, half of mi, vi, vi dash, whole square, where mi is the mass of the ith particle.

03:14 Okay? and in the arbitrary reference frame, we denote a quantity k of cm, which is the kinetic energy of the center of mass, which will be equal to half of m, capital m, vcm square.

03:32 Okay? where capital m is nothing but the summation of the individual particles in the nth particle system.

03:42 Okay? so, now, total kinetic energy, total kinetic energy in this arbitrary frame of reference, if we denote this quantity by k, then it will be equal to half of, since we are talking about the total quantity, this will be summation, i is equal to 1 to i is equal to n, half of mi, vi, whole square...

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