Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:$(i)$ Show $p=p_{i}^{\prime}+m_{i} V$ Where $p_{i}$ is the momentum of the $i^{\text {th }}$ particle (of mass $\left.m_{i}\right)$ and $p_{i}=m_{i} v_{i}^{t} .$ Note $v_{i}^{\prime}$ is the velocity of the $\mathrm{i}^{\mathrm{ith}}$ particle with respect to the centre of mass.Also, verify using the definition of the centre of mass that $\Sigma p_{i}=0$(ii) Prove that $\mathrm{K}=\mathrm{K}^{\prime}+1 / 2 \mathrm{MV}^{2}$ Where $K$ is the total kinetic energy of the system of particles, $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken relative to the center of mass and $\mathrm{MV}^{2} / 2$ is the kinetic energy of the translation of the system as a whole.

Home » Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass: Show Where is the momentum of the particle (of mass and Note is the velocity of the particle with respect to the centre of mass.Also, verify using the definition of the centre of mass that (ii) Prove that Where is the total kinetic energy of the system of particles, is the total kinetic energy of the system when the particle velocities are taken relative to the center of mass and is the kinetic energy of the translation of the system as a whole.

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
$(i)$ Show $p=p_{i}^{\prime}+m_{i} V$ Where $p_{i}$ is the momentum of the $i^{\text {th }}$ particle (of mass $\left.m_{i}\right)$ and $p_{i}=m_{i} v_{i}^{t} .$ Note $v_{i}^{\prime}$ is the velocity of the $\mathrm{i}^{\mathrm{ith}}$ particle with respect to the centre of mass.Also, verify using the definition of the centre of mass that $\Sigma p_{i}=0$
(ii) Prove that $\mathrm{K}=\mathrm{K}^{\prime}+1 / 2 \mathrm{MV}^{2}$ Where $K$ is the total kinetic energy of the system of particles, $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken relative to the center of mass and $\mathrm{MV}^{2} / 2$ is the kinetic energy of the translation of the system as a whole.

CBSE | Class 11 | NCERT | Physics | System of Particles and Rotational Motion

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
$(i)$ Show $p=p_{i}^{\prime}+m_{i} V$ Where $p_{i}$ is the momentum of the $i^{\text {th }}$ particle (of mass $\left.m_{i}\right)$ and $p_{i}=m_{i} v_{i}^{t} .$ Note $v_{i}^{\prime}$ is the velocity of the $\mathrm{i}^{\mathrm{ith}}$ particle with respect to the centre of mass.Also, verify using the definition of the centre of mass that $\Sigma p_{i}=0$
(ii) Prove that $\mathrm{K}=\mathrm{K}^{\prime}+1 / 2 \mathrm{MV}^{2}$ Where $K$ is the total kinetic energy of the system of particles, $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken relative to the center of mass and $\mathrm{MV}^{2} / 2$ is the kinetic energy of the translation of the system as a whole.

i)Here $\vec{r}_{i}=\vec{r}_{i}+\vec{R}+R \ldots$ also, $\vec{V}_{i}=\vec{V}_{i}+\vec{V} \ldots \ldots .$

Where $\vec{r}_{i}^{\overrightarrow{3}}$ and $\vec{v}_{i}^{\overrightarrow{3}}$ denote the radius vector and velocity of the $\mathrm{i}^{\text {th }}$ particle refers to the new origin as the centre of mass $\mathrm{O}$ ‘, and $\vec{V}$ is the velocity of the centre of mass with respect to $O$ .