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System of Particles and Rotational Motion

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Physics Part I

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Physics

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12

A small mass attached to a string rotats on frictionless table top as shown. If the tension is the string is increased by pulling the string causing the radius of the circular  motion to decrease by a factor of 2, the kinetic energy of the mass will

  • remain constant

  • increase by a factor of 2

  • increase by a factor 4 

  • decrease by a factor of 2


C.

increase by a factor 4 

From the law of conservation of angular momentum

mvr space equals space mv apostrophe straight r over 2
straight v apostrophe space equals space 2 straight v

So comma space

straight k subscript 1 over straight k subscript 2 space equals space fraction numerator begin display style 1 half end style mv squared over denominator begin display style 1 half end style mv apostrophe squared end fraction
straight k subscript 1 over straight k subscript 2 space equals space fraction numerator straight v squared over denominator straight v apostrophe squared end fraction

space equals space fraction numerator straight v squared over denominator left parenthesis 2 straight v right parenthesis squared end fraction

straight k subscript 1 over straight k subscript 2 space equals space 1 fourth

straight k subscript 1 space equals space 4 straight k subscript 1

From the law of conservation of angular momentum

mvr space equals space mv apostrophe straight r over 2
straight v apostrophe space equals space 2 straight v

So comma space

straight k subscript 1 over straight k subscript 2 space equals space fraction numerator begin display style 1 half end style mv squared over denominator begin display style 1 half end style mv apostrophe squared end fraction
straight k subscript 1 over straight k subscript 2 space equals space fraction numerator straight v squared over denominator straight v apostrophe squared end fraction

space equals space fraction numerator straight v squared over denominator left parenthesis 2 straight v right parenthesis squared end fraction

straight k subscript 1 over straight k subscript 2 space equals space 1 fourth

straight k subscript 1 space equals space 4 straight k subscript 1

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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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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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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. 
943 Views

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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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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