What is Centre of mass show that the Centre of mass of two particle system of equal masses lie at the Centre of line joining them?

Text Solution

Solution : Let `overset rarr(r_(1)), overset rarr(r_(2))` be the position vectors of two particle of masses `m_(1)` and `m_(2)` situated at `A` and `B` respectively. Let the origin `O` of the frame of reference coincide with the centre of mass of the two particles. <br> `:. m_(1) overset rarr(r_(1)) + m_(2) overset rarr(r_(2)) = 0 or overset rarr(r_(1)) = - (m_(2))/(m_(1)) overset rarr(r_(2))` <br> or `|overset rarr(r_(1))| = (m_(2))/(m_(1)) |overset rarr(r_(2))| or (r_(1))/(r_(2)) = (m_(2))/(m_(1))` <br> Which was to be proved. <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/PR_XI_V01_C05_S01_039_S01.png" width="80%">

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Answer

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Hint To answer this question we should be knowing the concept of centre of mass. The centre of mass is defined as the distribution of mass in the space in a unique point where the weighted relative position of the distributed mass sums to the value of zero.

Complete step by step answer

We know that the centre of mass of the two particles that is lying on the line joining the particles.Let us consider that the centre of mass lies at the point C.So, we can write the expression as follows$({m_1} + {m_2})x = {m_1}(0) + {m_2}(L)$So, the expression of x can be written as:$x = \dfrac{{{m_2}L}}{{{m_1} + {m_2}}}$So, we can say that the centre of mass of two particles lies on the line joining the particles.

Hence the correct answer is option A

Note We should know that the centre of mass is identified as the position which is relative to the position of the object or system of the objects. It is calculated as the simple average of the position of all the parts of the system, which is weighted according to their masses.

For simple rigid objects with the uniform density, the centre of mass is located at the centroid. The use of the centre of mass is to find the reference point for calculations in the mechanics that involves the masses which is distributed in the space, such as the linear and angular momentum of the planetary bodies and the rigid body dynamics.