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The centre of mass of a system of particles is at the origin. It follows that:
(A) The number of particles to the right of the origin is equal to the left of the origin
(B) The total mass of the particles to the right of the origin is the same as the total mass to the left of the origin
(C) The number of particles on the x-axis should be equal to the number of particles on the y-axis
(D) If there is a particle on the +ve X-axis, there should be at least one particle on the –ve X-axis
(E) None of these

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Last updated date: 22nd Jul 2024
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Answer
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Hint: In the given question, we have been given a statement that the centre of mass of a system of particles. We have been told that the centre of mass lies at the origin; we have now been asked to infer further information from this statement and state whether any of the situations given in the options might fulfil the condition of the statement. We will use the basic knowledge of the centre of mass to analyse this question.
Formula Used: \[{{X}_{CM}}={{m}_{1}}{{x}_{1}}+{{m}_{2}}{{x}_{2}}+----+{{m}_{n}}{{x}_{n}}\]

Complete step by step solution:
We know that the mathematical expression for the centre of mass can be given as \[{{X}_{CM}}={{m}_{1}}{{x}_{1}}+{{m}_{2}}{{x}_{2}}+----+{{m}_{n}}{{x}_{n}}\] where \[{{m}_{n}}\] is the mass of a given particle and \[{{x}_{n}}\] is its distance from the origin along the x-axis
For the centre of mass of a system of particles to be at origin \[{{X}_{CM}}\] must be zero, that is the combined sum of the product of a mass of the particle and its respective distance from the origin along the x-axis must be zero.
Let us now analyse the options given to us.
The first option simply states that the number of particles to the right of the origin is equal to the left of the origin, without giving us any information about the mass of the particles or their distances from the origin. Hence this information is inconclusive and incomplete and we cannot say if the centre of mass will or will not lie at the origin.
Similarly, the second option only provides information about the sum of the masses to the left and right of the origin and does not give any information about the distance of the masses. Since we are not just concerned with the sum of the masses but the sum of the product of the masses and their respective distance, we can say that the second option is also not correct as it does not provide any concrete evidence.
The third and the fourth options simply state information about the number of particles and do not tell us anything about the mass of the particles or their distances, and are hence inconclusive as well.
Since none of the first four options can surely lead to the conclusion given in the question,

we can say that option (E) is the correct answer.

Note:
In simple terms, we can say that the centre of mass of a system of particles lies at the origin if and only if the masses on either side of the y-axis or the x-axis are equal and are symmetrically distributed such that the sum of the product of their masses and their positions must furnish a zero value. Students often form the misconception that centre of mass only requires the criteria for mass but they should keep in mind that the distance of the mass also plays a pivotal role.