Centre of Mass Homework Help, Assignment Help, Online Tutor, Tutoring

Submit Assignment

* *

Payment Online Assignment Help

Homework Help

sfsf

Centre of Mass

The center-of-mass of an extended system of n-particles is defined as a point, whose position vector in an inertial frame S is given by

where r_i is the position vector of ith particle (mass m_i) and M is the total mass of the system.

Differentiating above eq. with respect to time, we define the velocity of the center-of-mass point:

Or, M V_CM = m₁ v₁ + m₂ v₂ + …. + m_n v_n

= p₁ + p₂ + …. + p_n = P_sys

Hence, we find P_sys = M V_CM = P_CM

That is, center-of-mass is a (hypothetical) point given by R_CM, where we can assume that the entire mass of the system is located, such that the total momentum of the extended system is equal to the momentum of its center-of-mass. That is, we have effectively reduced an extended object to a point mass M at R_CM. The center-of-mass is therefore also referred to as center-of-momentum.

For a centinuous distribution of mass points, the center-of-mass is given by,

where dm = ρ (r) dV is the mass of small volume dV located at position r; ρ (r) is the local density at r.