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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
Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is rwith components x, y, z and momentum is p with components px, py and pz. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.

straight I subscript straight x space equals space straight y space straight p subscript straight z space minus space straight z space straight p subscript straight y space
straight I subscript straight y space equals space zp subscript straight x space minus space xp subscript straight z space
straight I subscript straight z space equals space xp subscript straight y space minus space yp subscript straight x

Linear space momentum space of space
the space particle comma space straight p with rightwards harpoon with barb upwards on top equals space straight p subscript straight x space straight i with hat on top space plus space straight p subscript straight y space straight j with hat on top space plus space straight p subscript straight z space straight k with hat on top space space

Position space vector space of space
the space particle comma space straight r with rightwards harpoon with barb upwards on top space equals space straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top space

Angualr space momentum comma

space straight l with rightwards harpoon with barb upwards on top space equals space stack straight r space with rightwards harpoon with barb upwards on top space straight x space straight p with rightwards harpoon with barb upwards on top space

space space space space space equals space open parentheses straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top space close parentheses space straight x space space open parentheses straight p subscript straight x space straight i with hat on top space plus space straight p subscript straight y space straight j with hat on top space plus space straight p subscript straight z space straight k with hat on top close parentheses space

space space space space space equals space open vertical bar table row cell space straight i with hat on top end cell cell space straight j with hat on top space end cell cell straight k with hat on top space end cell row straight x straight y straight z row cell straight p subscript straight x end cell cell space straight p subscript straight y space end cell cell straight p subscript straight z end cell end table close vertical bar space

straight l subscript straight x space straight i with hat on top space plus space straight l subscript straight y space straight j with hat on top space plus space straight l subscript straight z space straight k with hat on top space equals space straight i with hat on top open parentheses space straight y space straight p subscript straight z space minus space straight z space straight p subscript straight y space close parentheses space minus space straight j with hat on top space left parenthesis zp subscript straight x space minus space xp subscript straight z space right parenthesis space plus space straight k with hat on top space left parenthesis xp subscript straight y space minus space yp subscript straight x right parenthesis space

Comparing space the space coefficients space of space space straight i with hat on top comma space space straight j with hat on top space space and space straight k with hat on top comma space we space get

right enclose straight l subscript straight x space equals space straight y space straight p subscript straight z space minus space straight z space straight p subscript straight y space
straight l subscript straight y space equals space zp subscript straight x space minus space xp subscript straight z
straight l subscript straight z space equals space xp subscript straight y space minus space yp subscript straight x end enclose space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... space left parenthesis straight i right parenthesis thin space

The space particle space moves space in space the space straight x minus straight y space plane. space

Hence comma space the space straight z space component space of space the space position space
vector space and space linear space momentum space vector space becomes space zero.

straight z space equals space straight p subscript straight z space equals space 0 space

Thus comma space eqution space left parenthesis straight i right parenthesis space reduces space to comma space

straight l subscript straight x space equals space 0
straight l subscript straight y space space equals space 0
straight l subscript straight z space equals space space xp subscript straight y space minus space yp subscript straight x

When the particle is confined to move in the x-y plane, the direction of angular momentum is along z direction. 

Hence, the result. 
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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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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 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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