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

Which of the following statements is false for the angular momentum $$\vec{L}$$ about the origin?

Coordinates of points:

$$A = \left( R\cos45^\circ, R\sin45^\circ \right) = \left( \frac{R}{\sqrt{2}}, \frac{R}{\sqrt{2}} \right)$$

$$B = \left( \frac{R}{\sqrt{2}} + a, \frac{R}{\sqrt{2}} \right)$$ (moving distance $a$ along the x-axis)

$$C = \left( \frac{R}{\sqrt{2}} + a, \frac{R}{\sqrt{2}} + a \right)$$ (moving distance $a$ along the y-axis)

$$D = \left( \frac{R}{\sqrt{2}}, \frac{R}{\sqrt{2}} + a \right)$$ (moving distance $a$ back along the x-axis)

The general formula for angular momentum is $$\vec{L} = \vec{r} \times \vec{p} = m(\vec{r} \times \vec{v})$$.

Path A to B (Moving Right)

Velocity: $$\vec{v} = v\hat{i}$$

Constant y-component: $$y = \frac{R}{\sqrt{2}}$$

$$\vec{L} = m \left( x\hat{i} + \frac{R}{\sqrt{2}}\hat{j} \right) \times (v\hat{i}) = -mv \frac{R}{\sqrt{2}}\hat{k}$$

Statement C is TRUE.

Path B to C (Moving Up)

Velocity: $$\vec{v} = v\hat{j}$$

Constant x-component: $$x = \frac{R}{\sqrt{2}} + a$$

$$\vec{L} = m \left( \left[\frac{R}{\sqrt{2}} + a\right]\hat{i} + y\hat{j} \right) \times (v\hat{j}) = mv \left( \frac{R}{\sqrt{2}} + a \right)\hat{k}$$

Statement A is TRUE.

Path C to D (Moving Left)

Velocity: $$\vec{v} = -v\hat{i}$$

Constant y-component: $$y = \frac{R}{\sqrt{2}} + a$$

$$\vec{L} = m \left( x\hat{i} + \left[\frac{R}{\sqrt{2}} + a\right]\hat{j} \right) \times (-v\hat{i}) = mv \left( \frac{R}{\sqrt{2}} + a \right)\hat{k}$$

Statement D is TRUE.

Path D to A (Moving Down)

Velocity: $$\vec{v} = -v\hat{j}$$

Constant x-component: $$x = \frac{R}{\sqrt{2}}$$

$$\vec{L} = m \left( \frac{R}{\sqrt{2}}\hat{i} + y\hat{j} \right) \times (-v\hat{j}) = -mv \frac{R}{\sqrt{2}}\hat{k}$$

 Statement B is FALSE because it incorrectly gives a positive $$+\hat{k}$$ direction.