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Illustration 11.4: Angular Momentum and Area

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One of the strangest ideas regarding angular momentum is that an object moving in a straight line can have angular momentum. Angular momentum for a particle is given by the cross product L = r × p. Given this, we see that the origin matters for the calculation of angular momentum for a particle.

In the absence of a net external torque acting on a system, a particle's angular momentum remains constant. For this discussion, the particle is free, so angular momentum should be conserved. Is there a different way to state the concept of angular momentum conservation? There may be. Consider the statement, Does a particle sweep out equal areas in equal times (with respect to any origin)?

Specifically, in this Illustration does a free particle moving in a straight line sweep out equal areas in equal times?

Press "start" to begin the animation, and let the show begin: A black dot will move freely from left to right. Different colors show the area the particle sweeps out with respect to some fixed point (the origin). Do all the areas have the same size? Click within each area and see what will happen. Certainly from mathematical equations we know the area of a triangle = width * height/2. All of the areas have the same height and the same width (= vx*dt).

Note: Kepler's second law (see Chapter 12 on gravitation for more details) states, during equal time intervals, the radius vector from the sun to a planet sweeps out equal areas. What does this tell you about the angular momentum of the planets?

Illustration authored by Fu-Kwun Hwang and Mario Belloni.
Applet authored by Fu-Kwun Hwang, National Taiwan Normal University.

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