Answer
64.8k+ views
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.
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.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)