A thin rod of length ‘L’ is lying along the x-axis with its ends at x = 0 and x = L. Its linear density (mass/length) varies with x as, where n can be zero or any positive number. If the position xCM of the centre of mass of the rod is plotted against ‘n’, which of the following graphs best approximates the dependence of xCM on n?  from Physics System of Particles and Rotational Motion Class 11 Manipur Board
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System of Particles and Rotational Motion

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Physics Part I

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A thin rod of length ‘L’ is lying along the x-axis with its ends at x = 0 and x = L. Its linear density (mass/length) varies with x as,straight k space open parentheses straight x over straight L close parentheses to the power of straight n where n can be zero or any positive number. If the position xCM of the centre of mass of the rod is plotted against ‘n’, which of the following graphs best approximates the dependence of xCM on n? 


A.

space straight x subscript cm space equals space fraction numerator integral dmx over denominator integral dm end fraction
fraction numerator integral begin display style λdx end style begin display style. end style begin display style straight x end style over denominator integral begin display style dm end style end fraction
space equals space fraction numerator integral begin display style straight k end style begin display style begin display style open parentheses straight x over straight L close parentheses end style squared end style begin display style. end style begin display style xdx end style over denominator straight k open parentheses begin display style straight x over straight L end style close parentheses to the power of straight n space dx end fraction
space equals space open square brackets fraction numerator begin display style fraction numerator kx to the power of straight n plus 2 end exponent over denominator left parenthesis straight n plus 2 right parenthesis straight L to the power of straight n end fraction end style over denominator begin display style fraction numerator kx to the power of straight n plus 1 end exponent over denominator left parenthesis straight n plus 1 right parenthesis straight L to the power of straight n end fraction end style end fraction close square brackets subscript 0 superscript straight L
space equals space open square brackets fraction numerator straight x left parenthesis straight n plus 1 right parenthesis over denominator straight n plus 2 end fraction close square brackets subscript 0 superscript straight L
straight x subscript cm space equals space straight L over 2 comma fraction numerator 2 straight L over denominator 3 end fraction comma fraction numerator 4 straight L over denominator 5 end fraction comma fraction numerator 5 straight L over denominator 6 end fraction comma.....
space straight x subscript cm space equals space fraction numerator integral dmx over denominator integral dm end fraction
fraction numerator integral begin display style λdx end style begin display style. end style begin display style straight x end style over denominator integral begin display style dm end style end fraction
space equals space fraction numerator integral begin display style straight k end style begin display style begin display style open parentheses straight x over straight L close parentheses end style squared end style begin display style. end style begin display style xdx end style over denominator straight k open parentheses begin display style straight x over straight L end style close parentheses to the power of straight n space dx end fraction
space equals space open square brackets fraction numerator begin display style fraction numerator kx to the power of straight n plus 2 end exponent over denominator left parenthesis straight n plus 2 right parenthesis straight L to the power of straight n end fraction end style over denominator begin display style fraction numerator kx to the power of straight n plus 1 end exponent over denominator left parenthesis straight n plus 1 right parenthesis straight L to the power of straight n end fraction end style end fraction close square brackets subscript 0 superscript straight L
space equals space open square brackets fraction numerator straight x left parenthesis straight n plus 1 right parenthesis over denominator straight n plus 2 end fraction close square brackets subscript 0 superscript straight L
straight x subscript cm space equals space straight L over 2 comma fraction numerator 2 straight L over denominator 3 end fraction comma fraction numerator 4 straight L over denominator 5 end fraction comma fraction numerator 5 straight L over denominator 6 end fraction comma.....
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Is it necessary for centre of mass to lie within the body?

No, centre of mass needs not to lie within the body. It is not necessary that the total mass of the system be actually present at the centre.

The position of the centre of mass is calculated using the usual Newtonian type of equations of motion. 
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What is the need of centre of mass?

Newton’s second law of motion is strictly applicable to point masses only. To apply the Newton's law of motion to rigid bodies, the concept of centre of mass is introduced.

The concept of centre of mass of a system enables us to discuss overall motion of the system by replacing the system by an equivalent single point object. 
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Is it necessary that there should be matter at the centre of mass of system?

No, it is not necessary that there be matter at the centre of mass of the system.

For e.g., if two equal point masses are separated by certain distance, the centre of mass lies at the mid point of two point masses and there is no mass at that point.
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Define centre of mass.

Centre of mass of a body or a system of bodies is a point at which the entire mass of the body or system is supposed to be concentrated. 
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What is the significance of defining the center of mass of a system?

The motion of n particle system can be reduced to one particle motion.

An equivalent single point object would enable us to discuss the overall motion of the system. 
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