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Establish relationship between torque an...

Establish relationship between torque and angular momentum.

Text Solution

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We know that, angular momentum is given by
`vecL=vecrxxvecp`
Differentiating w.r.t. t, we get
`(dvecL)/(dt)=d/(dt)(vecrxxvecp)=(dvecr)/(dt)xxvecpxxvecr=(dvecp)/(dt)`
But `(dvecr)/(dt) = vecv and (dvecp)/(dt) = vecF`
`:. (dvecL)/(dt) = vecv xxmvecv xxvecrxxvecF=mvecvxxvecv +vecrxxvecF`
But `vecvxxvecv=0" "` ( `:.` cross product of parallel vectors is zero)
`implies(dvecL)/(dt) = vecrxxvecF=vectau" "[:. vectau=vecrxxvecF]`
Hence `(dvecL)/(dt)=vectau`
Thus, torque acting on a particle is equal to rate of change of angular momentum.
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