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A particle of mass m is moving along the side of a square of side a, with a uniform speed v in the XY-plane as shown in the figure

Which of the following statements is false for the angular momentum L about the origin?

A) $L= -\frac{-mv}{\sqrt{2}}R\widehat{k}$ when the particle is moving from A to B

B) $L=mv (\frac{R}{\sqrt{2}}-a)\widehat{k}$ when the particle is movinf from C to D

C) $L=mv (\frac{R}{\sqrt{2}}+a)\widehat{k}$ when the particle is moving from B to C

D) $L= -\frac{mv}{\sqrt{2}}R\widehat{k}$ when the particle is moving from D to A

Option B,D

We can apply L=m(r × v) for different parts

For example

if part (a) , cooredinates of A are $(\frac{R}{\sqrt{2}},\frac{R}{\sqrt{2}})$

there fore ,r = $\frac{R}{\sqrt{2}}\widehat{i}+\frac{R}{\sqrt{2}}\widehat{j}$ and $v=v\widehat{i}$

So, subsuituting in L=m(r× v) we get,

$L= -\frac{mvR}{\sqrt{2}}\widehat{R}$

Hence, option (a) is correct, similarly , we can check other options also

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