The value of α for which angular momentum about origin is conserved is 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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Physics

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straight A space force space straight F space equals space straight alpha stack space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top space is space acting space at space straight a space point space straight r equals space 2 straight i with hat on top space minus space 6 space straight j with hat on top space minus 12 space straight k with hat on top. The value of α for which angular momentum about origin is conserved is
  • -1

  • 2

  • zero

  • 1


A.

-1

When the resultant external torque acting on a system is zero, the total angular momentum of a system is zero, the total angular momentum of a system remains constant.This is the principle of the conservation of angular momentum.

Given,

straight F space equals space straight alpha space straight i with hat on top space plus 3 space straight j with hat on top space plus space 6 space straight k with hat on top space is space acting space at space straight a space point space straight r equals space 2 space straight i with hat on top space minus 6 space straight j with hat on top space minus 12 space straight k with hat on top
As comma space angular space momentum space about space origin space is space consverved
straight i. straight e. comma
Torque space equals space constant
rightwards double arrow space Torque space equals space 0 space rightwards double arrow space straight r space xF space equals space 0

open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 2 cell negative 6 end cell cell negative 12 end cell row straight alpha 3 6 end table close vertical bar space equals space 0

rightwards double arrow space left parenthesis negative 36 space plus 36 right parenthesis straight i with hat on top space minus space left parenthesis 12 space plus space 12 straight alpha right parenthesis space straight j with hat on top space plus left parenthesis 6 space plus 6 straight alpha right parenthesis space straight k with hat on top space equals 0

rightwards double arrow space 0 space straight i with hat on top space minus space 12 space straight i with hat on top space left parenthesis space 1 plus straight alpha right parenthesis straight i with hat on top space plus space 6 space left parenthesis 1 space plus space straight alpha right parenthesis straight k with hat on top space equals 0

6 space left parenthesis 1 space plus straight alpha right parenthesis space equals 0 space rightwards double arrow space straight alpha space equals negative 1

When the resultant external torque acting on a system is zero, the total angular momentum of a system is zero, the total angular momentum of a system remains constant.This is the principle of the conservation of angular momentum.

Given,

straight F space equals space straight alpha space straight i with hat on top space plus 3 space straight j with hat on top space plus space 6 space straight k with hat on top space is space acting space at space straight a space point space straight r equals space 2 space straight i with hat on top space minus 6 space straight j with hat on top space minus 12 space straight k with hat on top
As comma space angular space momentum space about space origin space is space consverved
straight i. straight e. comma
Torque space equals space constant
rightwards double arrow space Torque space equals space 0 space rightwards double arrow space straight r space xF space equals space 0

open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 2 cell negative 6 end cell cell negative 12 end cell row straight alpha 3 6 end table close vertical bar space equals space 0

rightwards double arrow space left parenthesis negative 36 space plus 36 right parenthesis straight i with hat on top space minus space left parenthesis 12 space plus space 12 straight alpha right parenthesis space straight j with hat on top space plus left parenthesis 6 space plus 6 straight alpha right parenthesis space straight k with hat on top space equals 0

rightwards double arrow space 0 space straight i with hat on top space minus space 12 space straight i with hat on top space left parenthesis space 1 plus straight alpha right parenthesis straight i with hat on top space plus space 6 space left parenthesis 1 space plus space straight alpha right parenthesis straight k with hat on top space equals 0

6 space left parenthesis 1 space plus straight alpha right parenthesis space equals 0 space rightwards double arrow space straight alpha space equals negative 1

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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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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 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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