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State and Prove : Law of Conservation of Angular Momentum - Physics

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Derivation

State and prove : Law of conservation of angular momentum.

State and prove principle of conservation of angular momentum

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Solution 1

Statement:-
The angular momentum of a body remains constant, if resultant external torque acting on the body is zero.

Proof:-
a. Consider a particle of mass m, rotating about an axis with torque ‘τ’.

Let `vecp` be the linear momentum of the particle and `vecr` be its position vector.

b. By definition, angular momentum is given by, `vecL=vecrxxvecp`                                 ....................(1)

c. Differentiating equation (1) with respect to time t, we get,

`vec(dL)/dt=d/dt(vecrxxvecp)`

`thereforevec(dL)/dt=vecrxxvec(dp)/dt+vecpxxvec(dr)/dt` ..................(2)

d. 

`"But,"vec(dr)/dt=vec"v",vec(dp)/dt=vecF" and "vecp="mv"`

∴ Equation (2) becomes,

`vec(dL)/dt=vecrxxvecF+0`                                            [`becausevec"v"xxvec"v"="v"^2sin0^@=0`]

e. `"Also, "vectau=vecrxxvecF`

`thereforevec(dL)/dt=vectau`

f. If resultant external torque (τ) acting on the particle is zero, then `vec(dL)/dt=0.`

`thereforevecL="constant"`

Hence, angular momentum remains conserved.

Solution 2

Principle (or law) of conservation of a body is conserved if the resultant external torque on the body is zero.
Proof: Consider a particle of mass m whose position vector with respect to the origin at any instant is `vecr`

Then, at this instant, the linear velocity of this particle is `vecv = vec(dr)/(dt)`, its linear momentum is `vecp = mvecv` and its angular momentum about an axis through the origin is `vecl = vecr xx vecp`

Its angular momentum `vecl` may change with time due to a torque on the particle.

`vec(dl)/(dt) = d/(dt) (vecr xx vecp)`

`=(dvecr)/(dt) xxvecp + vecr xx (dvecp)/(dt)`

`=vecv xx vec(mv) + vecr + vecF`

`=vecr xx vecF`

`= vecr xx vecF`       (∵ `vecv xxvec v = 0`)

= `vect`

Where = `(dvecp)/(dt) =  vecF`, the force on the particle.

Hence if `vect = 0, vec(dl)/(dt) = 0`

∴`vecl` = constant, i.e `vecl` is conserved. This proves the principle (or law) of convervation of angualar momentum.

Concept: Angular Momentum or Moment of Linear Momentum
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