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I'm taking AP Physics C and right now in our rotation unit I am confused about why angular momentum is constant in certain cases.

The situation involves the angular momentum of an object moving at constant velocity in a straight line that does not pass through the origin.

Why is it constant? While the linear velocity is constant, the angular velocity of the particle relative to a circle centered on the origin, and a radius such that the point is on the circumference changes as the particle moves. If the particle had an extremely high x, wouldn't most of the velocity vector point towards the circle instead of tangent to it, there by making mot of the linear velocity not angular velocity?

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  • $\begingroup$ It is just a consequence of the mathematical definitions. The angular momentum is radius vector multiplied by the linear momentum and the sine of the angle those two make with each other. You know the linear momentum is conserved i.e. constant. The sine multipled by the radius vector will make the perpendicular distance, and this distance is constant no matter how far away the particle moved. $\endgroup$ Commented Apr 23 at 16:00

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By the definition $$\mathbf{L} = \color{blue}{\mathbf{r}} \times \color{red}{\mathbf{p}} = \hat{\mathbf{n}}|\color{red}{\mathbf{p}}||\color{blue}{\mathbf{r}}|\sin\theta$$ we see that $|\color{blue}{\mathbf{r}}|\sin\theta$, the vertical length, is constant. Since $\color{red}{\mathbf{p}}$ and $\hat{\mathbf{n}}$ are constant, so is $\mathbf{L}$.

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Another way to state the definition of angular momentum about a given axis for a point particle: it is the product of linear momentum and perpendicular distance from the given axis to the line along which the particle is travelling.

If the momentum is constant then neither of those quantities is changing.

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naturallyInconsistent's comment has helped me understand why this is true. Angular momentum is not just defined as the product, but the cross product of the point's linear momentum vector and radius vector. Cross products multiply the 2 values together, then by the sine of the angle between them. Eg.

$$A\times B=A*B*sin(\theta)$$

The angle between the radius and linear momentum is the same as the angle between the radius and the "floor" through the origin. The sine of this angle multiplied by the radius vector will always give the vertical portion of the radius vector. As this particle is only moving horizontally, the height of the radius vector remains constant. In the situation defined above the linear momentum is constant. This means that both components of angular momentum in this situation are constant.

Here is a desmos graph illustrating how the vertical part of the radius, in green, stays constant as the point moves horizontally.

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