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Principle of Work/Energy Plane Motion

Key Concept: The principle of work and energy comes from integration of Euler's equations of

motion for a rigid body undergoing translation and rotation in a plane.

In a Nut Shell: The principle of work and energy in a plane is that the work done on the body or bodies from position 1 to position 2 equals the change in kinetic energy of the body or bodies from position 1 to position 2.

W_1-2= T₂ ˗ T₁

Principle of Work/Energy

where

W_1-2 = the work by all forces and moments acting on the body or bodies that actually

do work on the body or bodies from position 1 to position 2

T₂ = the total kinetic energy of the body or bodies in position 2

T₁ = the total kinetic energy of the body or bodies in position 1

The key steps in applying the Principle of Work and Energy in plane motion are as follows:

Step 1: Draw a free body diagram showing all external forces and moments acting on the body or bodies. Note that not all forces acting on the body or bodies may actually do work.

Step 2: Calculate the work done in going from position 1 to position 2 by those forces and moments that actually do work. Common forces include those due to gravity, applied loads, friction, spring loading, and applied moments.

Step 3: Calculate the kinetic energy of the body or bodies in position 1 and in position 2. Note for plane motion a body may be both translating and rotating. So the total kinetic energy contains the kinetic energy “of” its center of mass (translation) plus the kinetic energy “about” its center of mass (rotation).

Note, you may need to include kinematics to relate the linear and angular velocities involved in the motions of one or more bodies that contribute to the total kinetic energy.

Total Kinetic Energy: T = ½ mv_c² + ½ I^c_zz ω² (ft lb or N m)

where ½ mv_c² = kinetic energy of the center of mass of each body; v_c = speed of mass center

½ I^c_zz ω² = kinetic energy of rotation for each body; ω = angular speed of body

m = mass of body I^c_zz = mass moment of inertia about the bodies center of mass

Work done by a spring W_1-2= ½ k ( d_i² – d_f²) (ft lb or N m)

where k = spring constant, (lb/ft, N/m) d_i = the initial extension of the spring, (ft, m) and

d_f = the final extension of the spring (ft, m)

Click here for examples.

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