Ch 7. Rigid Body Energy Methods Multimedia Engineering Dynamics Rot. Work & Energy Conservation of Energy
 Chapter - Particle - 1. General Motion 2. Force & Accel. 3. Energy 4. Momentum - Rigid Body - 5. General Motion 6. Force & Accel. 7. Energy 8. Momentum 9. 3-D Motion 10. Vibrations Appendix Basic Math Units Basic Equations Sections Search eBooks Dynamics Fluids Math Mechanics Statics Thermodynamics Author(s): Kurt Gramoll ©Kurt Gramoll

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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