Angular Momentum
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[edit] Definition
- L = r × mv
where L is the angular momentum, r is the position vector of the particle relative to the given point, m is the mass of the particle, v is the velocity of the particle, and × denotes the cross product operation on vectors.
If we denote the linear momentum by p, then we can write this simply as:
- L = r × p
You will notice that there is no such thing as the angular momentum of a particle, only its angular momentum relative to a given point. L will take on different values depending on which point we choose as the origin of our coordinate system.
[edit] The angular momentum of a system
Above, we defined the angular momentum of a particle with respect to some given point. It is also possible to define the angular momentum of a system of particles with respect to some given point: it is simply the sum of the angular momenta of the particles that make up the system.
[edit] Conservation of angular momentum
The law of conservation of angular momentum may be stated as follows:
- If a system of particles has no external forces acting on it, then its angular momentum with respect to a given point will be constant over time.
This constant will, of course, be different depending on our choice of origin for our coordinate system, but for any particular point, it will be constant. This can be a great convenience, since if we wish to exploit the law of conservation of angular momentum to analyze some physical system, then we can choose our point of origin to be whatever makes the mathematics easiest (see the example of the gyroscope given below, where we take the origin to be the center of the disc of the gyroscope.).
Note that the law refers to closed systems, not to the individual bodies contained in them. A body within such a system may undergo a change to its angular momentum, and all the conservation law has to tell us is that some other body or bodies in the system must also undergo a change in angular momentum such that L is conserved.
We shall give some simple examples of the conservation of angular momentum.
[edit] Example: motion in a straight line
Consider a particle of constant mass m which starts at position r0 moving with velocity v and having no forces acting on it. By Newton's First Law of Motion, the absence of forces acting on it means that it must continue to move in a straight line at the same speed, so its equation of motion is given by
- r = r0 + vt.
Its linear momentum is mv, so its angular momentum relative to the origin is given by
- L = r × mv = (r0 + vt) × mv.
The cross product is distributive over addition, so we can rewrite this as
- L = (r0 × mv) + (vt × mv)
Clearly the vectors vt and mv are parallel, since they are both in the direction of v; and the cross product of two parallel vectors is 0. So the last term in the sum above comes to zero, and we can write:
- L = r0 × mv
Now r0, m, and v are all constants in this system, so it follows that L is also constant, as required by the law of conservation of angular momentum.
[edit] Example: the gyroscope
Consider an idealized gyroscope, in which the mass of the spindle and the thickness of the disc are negligible. Suppose it is spinning and that no outside forces are acting on it. If we take our point of origin to be the center of the disc of the gyroscope, and consider the points on the disc as a system of particles, we find that every particle has a position vector in the plane in which the gyroscope is spinning, and a linear momentum also in the plane in which the gyroscope is spinning (and at right angles to its position vector).
Now if two vectors lie in the same plane and are not parallel, then their cross product is perpendicular to the plane they lie in. The angular momentum of a system is the sum of the angular momenta of its parts, so the angular momentum of the gyroscope must be also perpendicular to its plane of rotation, along the direction of its axis.
This means that for the gyroscope to topple over, that is, to change the direction of its axis, it would have to change its angular momentum. But the conservation law tells us that this is impossible without the action of an outside force. Therefore, in the absence of such a force, the gyroscope must go on rotating in the same plane.
In real life, there are forces acting on gyroscopes: friction, which causes them to lose speed, and gravity, which causes them to precess (wobble).
[edit] Misconceptions
One misconception about angular momentum is that it applies only to things which are rotating or orbiting or generally going round and round. We trust that the definition of angular momentum given, and the example of motion in a straight line, will have cured the reader of any such misapprehension.
Creationist misuses of the concept of angular momentum as it applies to the formation of the Solar System will be discussed in the following articles:
