Angular momentum is minimum when the angle between position vector and linear momentum is ..... degree

Understanding Angular Momentum

Angular momentum ($\mathbf{L}$) is given by the cross product of the position vector ($\mathbf{r}$) and the linear momentum ($\mathbf{p}$):

$\mathbf{L} = \mathbf{r} \times \mathbf{p}$

Analyzing the Cross Product

1. Cross Product Properties: The magnitude of the angular momentum vector is dependent on the sine of the angle ($\theta$) between the position vector and linear momentum:

   $|\mathbf{L}| = |\mathbf{r}||\mathbf{p}| \sin(\theta)$

2. Minimum Angular Momentum Condition: To minimize angular momentum, we want $|\mathbf{L}|$ to be as small as possible. The sine function ($\sin(\theta)$) reaches its minimum value of 0 when:

   $\sin(\theta) = 0 \implies \theta = 0^\circ \text{ or } 180^\circ$

Conclusion

Thus, the angle between the position vector and linear momentum for minimum angular momentum is either $0^\circ$ or $180^\circ$. This indicates that the position vector and momentum are aligned (pointing in the same or opposite directions), resulting in no angular momentum.

Final Answer

Angular momentum is minimum when the angle between position vector and linear momentum is $0^\circ$ or $180^\circ$.