Motion Of The Centre Of Mass - Practice Questions & MCQ

1. Velocity of the centre of mass

$
\vec{v}_{C M}=\frac{m_1 \overrightarrow{v_1}+m_2 \vec{v}_2 \ldots \ldots \ldots}{m_1+m_2 \ldots \ldots}
$

$\mathrm{m}_1, \mathrm{~m}_2----$ are mass of all the particles $\overrightarrow{v_1}, \vec{v}_2 \ldots \ldots$ are velocities of all the particles.
Similarly momentum of the system $=P_{\text {sys }}=M v_{c m}$

2. Acceleration of centre of mass

$
\vec{a}_{C M}=\frac{m_1 \overrightarrow{a_1}+m_2 \vec{a}_2 \ldots \ldots \ldots}{m_1+m_2 \ldots \ldots}
$

$\mathrm{m}_1, \mathrm{~m}_2$ are mass of all the particles $\overrightarrow{a_1}, \overrightarrow{a_2} \cdots$ are their respective acceleration.
Similarly Net force on the system $=F_{\text {net }}=M a_{c m}$
And $F_{n e t}=\overrightarrow{F_{e x t}}+\overrightarrow{F_{\text {int }}}$

And we know that both the action and reaction of an internal force must be within the system. In this way, vector summation will cancel all internal forces and hence net internal force on the system is zero.

So  $\overrightarrow{F_{\text {net }}}=M \overrightarrow{a_{c m}}$

3. If External Force = 0

$
\vec{F}_{e x t}=0 \Rightarrow M \vec{a}_{c m}=0 \Rightarrow \vec{a}_{c m}=0
$

if $\vec{a}_{c m}=0 \Rightarrow v_{c m}=$ constant
${ }_{\text {If }} v_{c m}=$ constant $\Rightarrow P_{\text {sys }}=$ constant

So it implies that the total momentum of the system must remain constant.

i.e. if no external force is acting on the system, the net momentum of the system remains constant. This is nothing but the principle of conservation of momentum in absence of external forces. Which says ìf resultant external force is zero on the system, then the net momentum of the system must remain constant.

• Special case

If External Force = 0 and Velocity of Centre of Mass = 0

Then centre of mass remains at rest. Individual components of a system may move and have non zero momentum due to  mutual forces but the net momentum of the system remains zero.