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DYNAMICS - THEORY




Angular Momentum Vector

 

Angular momentum is analogous to the principle of impulse and momentum. The angular momentum about a point (generally the origin, O) is

 
Ho = r × mv
 

The distance vector, r, is from the point of rotation to the object. This cross product will produce a vector, H, that is perpendicular to both r and v, as shown in the diagrams at the left.

Most problems deal with the angular momentum for an object moving in the x-y plane (about the origin O). Then the angular momentum equation simplifies to

     Ho = m|r||v| sinθ ez

where r is the position vector of the object and ez is the unit vector in the z direction.

The relationship between the angular momentum of an object and the external moment applied to the object can be determined by examining the cross product of the position vector and Newton's Second Law, or

     r × ΣF = r × ma = r × m dv/dt

The right-hand side of the this equation equals the time derivative of the angular momentum,

     d(r × mv)/dt = (dr/dt × mv) + (r × m dv/dt)

                       = (v × mv) + (r × m dv/dt)

                       = r × m dv/dt

Thus, the moment of external forces about O equals the object's rate of change of angular momentum about O:

     

Integrating this equation with respect to time gives

 
 

If the moment acting on an object about a fixed point O is known, it can be integrated to determine the object's change in angular momentum.

If the moment is zero, Ho is constant, i.e. angular momentum is conserved.

     
    Central-Force Motion

   

If the total force acting on the object remains directed towards a fixed point, such as the origin O, then the fixed point is called the center of motion, and the object is said to be in central-force motion.

The position vector r is always parallel to the force, thus the moment is zero:

     r × ΣF = 0 

Hence, for central-force motion, angular momentum is conserved, i.e. Ho is constant.

     
    Plane Central-Force Motion


Central-Force Motion

 

 

If an object undergoes central-force motion in a plane, then it is said to be in plane central-force motion.

If r and v are expressed in cylindrical coordinates then

     r = rer                     v = vrer + vθeθ

Substituting these into the equation for angular momentum gives

     Ho = r × mv

         = rer × m (vrer + vθeθ)

         = mr vθez

This equation indicates that for plane central-force motion, the product of the radial distance from the center of motion and the transverse component of velocity is constant, i.e. r vθ is constant.

     
   
 
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