Motion of Center of Mass
Consider a system of multiple particles. Each particle of that system is moving at a different velocity. How would someone assign a velocity to the system as a whole? Let us consider a system of particles m1, m2, m3 …and so on. The initial position vectors of these particles are r1, r2, r3 …rn. Now, these particles start moving in the directions of their position vectors. The goal is to find the velocity and the direction of the velocity of the Center of mass of the system.
From the definition of the Center of mass,
Mrcm = m1r1 + m2r2 + ….
Since the particles are in motion, they are changing their position vectors. Differentiating the equation from both sides.
[Tex]\frac{d}{dr}(M\vec{r}) = \frac{d}{dr}(m_1\vec{r}_1 + m_1\vec{r}_2 + m_1\vec{r}_3 + ….m_1\vec{r}_n)[/Tex]
⇒ [Tex]M\frac{d}{dr}(\vec{r}) = m_1\frac{d}{dr}(\vec{r}_1) + m_2\frac{d}{dr}(\vec{r}_2) + m_3\frac{d}{dr}\vec{r}_3 + ….[/Tex]
⇒ [Tex]M\vec{v} = m_1\vec{v_1} + m_2\vec{v_2} + m_3\vec{v_3}+ ….[/Tex]
⇒[Tex]\vec{v} = \frac{m_1\vec{v_1} + m_2\vec{v_2} + m_3\vec{v_3}+ ….}{M}[/Tex]
Similarly, if the particles are under acceleration. The equation given above can be differentiated again to find the acceleration of the Center of mass of the body.
[Tex]\vec{a} = \frac{m_1\vec{a_1} + m_2\vec{a_2} + m_3\vec{a_3}+ ….}{M}[/Tex]
Motion of Center of Mass
Center of Mass is an important property of any rigid body system. Usually, these systems contain more than one particle. It becomes essential to analyze these systems as a whole. To perform calculations of mechanics, these bodies must be considered as a single-point mass. The Center of mass denotes such a point. Often the mechanical systems move in a transitory or a rotatory manner. In that case, the Center of mass also moves and acquires some velocity and acceleration. Let’s see how to calculate these metrics for such systems in detail.