A particle of mass of m is moving along the side of square of side 'a' with a uniform speed u in the X-Y plane as shown in the figure.

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

Options

A. `vecL=-(mv)/(sqrt(2))R hatk` when the particle is moving from A to B.
B. `vecL= mv [((R)/(sqrt(2))-a] hatk` when the particle is moving from C to D.
C. `vecL= mv [(R/(sqrt(2))+a] hatk` when the particle is moving from B to C.
D. `vecL=(mv)/(sqrt(2))R hatk` when the particle is moving from D to A.

Answer

D

Explanation

Angular momentum about point "O" is
`vecL=m(vecrxxvecv)`
`=mrvsinthetahatn`
r is the perpendicular distance from axis of rotation to the velocity of particle.
We can find the direction of L by using method of cross product of vectors (right-hand rule) for each segment of square.

Along CD, distance from line of motion r `r=((R)/(sqrt2)+a)`
`:.vecL=((R)/(sqrt2)+a)mvhatk:.` option (b) is incorrect.
Moving along D to A, `r=((R)/(sqrt2)) but direction of L is in negative Z axis.
`:.vecL=(R)/(sqrt2)mv(-hatk):.` option (d) is incorrect.