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

Derivation

State and prove : Law of conservation of angular momentum.

State and prove principle of conservation of angular momentum

#### 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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