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DYNAMICS - THEORY

    General Work


Energy Balance

Force
Moment
Gravity
Friction
Spring
Rigid Body Work - Energy Terms
 

The Principle of Work and Energy equates the total work performed on a body to the change in kinetic energy,

     Σ Work = Δ Kinetic Energy

Another way to express this concept is

 
T1 + ΣU1-2 = T2
 

where
     T1 = kinetic energy (translational and rotational)
                at position 1
     T2 = kinetic energy (translational and rotational)
                at position 2
     ΣU1-2 = total change in energy (i.e. work)
                   between positions 1 and 2

Work on a rigid body is the same as work on a particle, with the addition of rotational energy. Recall from Particle Energy Methods section, there are various ways to model energy for a particle. These are listed in the table at the left.

     

Rotating Rigid Body
  Work of a Force Couple, or Moment

 

The work of two parallel, non-collinear forces in opposite directions with equal magnitudes forms a couple and moment. This moment generate energy similar to a linear force through a distance. However, the distance is now the angle of rotation.

 
 
     
    Kinetic Energy for a Rotating Body


Rotational and Translational
Kinetic Energy

 

For a particle that has no rotation, the kinetic energy is simply,

     T = 1/2 m v2

This equation can be applied to every particle that makes up a rigid body,

      

However, if the particle is a rigid body with dimensions, then the velocity needs to be written as

     

where G is the center of gravity. The second terms models the rotation of the object. Substituting vi into the kinetic equation and noting that a magnitude of vi/G = ri/G ω gives

 
T = 1/2 m vG2 + 1/2 IG ω2
 

If the rigid body has a fixed point, then the equation becomes

     T = 1/2 Io ω2

where Io is moment of inertia about the fixed point.

     
   
 
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