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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

Point masses m1 and m2 are placed at the opposite ends of a rigid rod of length L and negligible mass. The rod of length L and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity straight omega subscript straight o  is minimum is given by 

  • straight x space equals space fraction numerator straight m subscript 1 straight L over denominator straight m subscript 1 plus straight m subscript 2 end fraction
  • straight x space equals space straight m subscript 1 over straight m subscript 2 straight L
  • straight x space equals space straight m subscript 2 over straight m subscript 1 straight L
  • straight x space equals space fraction numerator straight m subscript 2 straight L over denominator straight m subscript 1 space plus straight m subscript 2 end fraction

D.

straight x space equals space fraction numerator straight m subscript 2 straight L over denominator straight m subscript 1 space plus straight m subscript 2 end fraction

As two point masses m1 and m2 are placed at opposite ends of a rigid rod of length L and negligible as shown in the figure. 


Total moment of inertia of the rod

 I = m1x2  + m2(L-x)2

I = m1x2 + m2L2 +m2x2 - 2m2Lx

As I is minimum i.e 

dI over dx space equals space 2 straight m subscript 1 space straight x space plus space space 0 space plus space 2 xm subscript 2 space minus space 2 straight m subscript 2 straight L space equals space 0

rightwards double arrow space straight x space left parenthesis 2 straight m subscript 1 space plus space 2 straight m subscript 2 right parenthesis space equals space 2 straight m subscript 2 straight L

rightwards double arrow space straight x space equals space fraction numerator straight m subscript 2 straight L over denominator straight m subscript 1 plus straight m subscript 2 end fraction
When I is minimum, the work done by rotating a rod will1 divided by 2 straight omega squared space with space angular space velocity space straight omega subscript straight o be minimum.


As two point masses m1 and m2 are placed at opposite ends of a rigid rod of length L and negligible as shown in the figure. 


Total moment of inertia of the rod

 I = m1x2  + m2(L-x)2

I = m1x2 + m2L2 +m2x2 - 2m2Lx

As I is minimum i.e 

dI over dx space equals space 2 straight m subscript 1 space straight x space plus space space 0 space plus space 2 xm subscript 2 space minus space 2 straight m subscript 2 straight L space equals space 0

rightwards double arrow space straight x space left parenthesis 2 straight m subscript 1 space plus space 2 straight m subscript 2 right parenthesis space equals space 2 straight m subscript 2 straight L

rightwards double arrow space straight x space equals space fraction numerator straight m subscript 2 straight L over denominator straight m subscript 1 plus straight m subscript 2 end fraction
When I is minimum, the work done by rotating a rod will1 divided by 2 straight omega squared space with space angular space velocity space straight omega subscript straight o be minimum.


2889 Views

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. 
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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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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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