# Angular Momentum

 Q:     In the earth-moon system the moon does not follow a true circle around the earth but the two bodies are said to dance around the system's center of mass.   Why do they dance?  Do the sun and the earth perform a similar dance? A:     Any complex system consisting of a number of objects is characterized by a "center of mass", a point in which the entire mass of the system seems to be concentrated.   The center of mass may be a point in empty space that does not coincide with the position of any mass-component of the system.  For example, the center of mass of a bicycle wheel is in the center of the wheel although all the masses are distributed along the rim.  The center of mass of a galaxy may be positioned in an empty point in space.  In the case of the earth-moon system, like in the case of a diatomic molecule, the center of mass is positioned somewhere on the straight line that connects the two masses (earth and moon or the two atoms).  The motion of the two objects, as it occurs in reality, can be described theoretically (mathematically) in any system of reference.  The system of reference: Cartesian, polar, etc., is the choice of the scientist who develops the theory.  In different systems of reference, the equations of the model are different although the physical phenomenon (motion of the considered objects) is the same.  If the origin of the system of reference is chosen at the location of one object, then the model should describe this object at rest and the other object in motion.  If the origin of the system of reference is chosen in the center of mass of the system, then the model should describe both objects in motion relative to the center of mass.  This appears as the "dance" mentioned in the question. Q:     How can a gyroscope be used for navigation? A:     A gyroscope is a very well "balanced" rotating object, meaning an object that is almost perfectly symmetric and homogeneous, rotating around an axis passing through its center of mass (which is at the same time its center of symmetry.)  For the rotating gyroscope, the angular momentum (L) is a constant vector quantity, meaning constant in magnitude and constant in direction, even if the fixture of the gyroscope or any object rigidly connected to this fixture (such as the body of an airplane or a ship) changes orientation is space.  When the body of an airplane, for example, deviates from a previously set course, the angle between the direction of motion of the aircraft and the axis of rotation of the gyroscope will change.  A computer will sense this change and will trigger an adjustment of course.

## References

### Equations

Rolling Bodies
 v cm = R
Torque as a Vector
 = r x F = rF sin = r F = r F
Angular Momentum of a particle
 l = r x p = m ( r x p ) l = rmv sin = r p = r p = r mv
Newton's Second Law in Angular Form

Angular Momentum of  System of Particles
Angular Momentum of a Rigid Body L = I
Conservation of Angular Momentum
 L = a constant Li =Lf
Quantized Angular Momentum S = ms  (h / 2 )

# Mechanics List of Topics

 Measurements Newton's Laws Potential Energy and Conservation of Energy Rotation of Rigid Bodies Elasticity Vectors Forces and Fields Linear Momentum Angular Momentum Mechanical Oscillations Motion of Point-Mass Objects in One Dimension The Gravitational Field Collisions Torque Mechanical Waves Motion of Point-Mass Objects in Two and Three Dimensions Kinetic Energy and Work Circular Motion of Point-Mass Objects Equilibrium Sound

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