Page 345 - Dynamics of Mechanical Systems

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326 Dynamics of Mechanical Systems

θ
d F
d/2
F
B
F

FIGURE 10.3.1 FIGURE 10.3.2
A couple applied to a flywheel. Displacement at the point of application of the force.

10.3 Work Done by a Couple
Suppose a couple is applied to a body B, causing B to rotate. Then, from our intuitive
understanding of work, as an exertion creating a movement, we expect that work has
been performed by the couple. To develop and quantify this, consider first a flywheel B
supported by a frictionless pin at its center. Let a couple C consisting of two equal
magnitude, but oppositely directed, forces be applied to B as in Figure 10.3.1. If B is initially
at rest, the application of the couple will cause B to rotate about its pin. Specifically, the
points of application of the forces will experience displacements with the forces, thus
doing work.
Let the forces of C change orientation as B rotates (that is, let C “rotate” with B). Consider
the work of one of these forces: from Figure 10.3.2, we see that the displacement at the
point of application of the force is (d/2)θ where d is the distance between the couple forces
and θ is the rotation angle. Then, from our definition of work of Eq. (10.2.2) we see that
the work done by one of the couple forces is:

W = ( ) 2 θ (10.3.1)
d
F
Hence, the work done by the couple is:

θ
W = F d = Tθ (10.3.2)

where T is the torque of the couple (see Section 6.4).
Consider now a generalization of this example where a couple C is applied to a body
B with B instantaneously rotating through a differential angle dθ about an axis parallel to
a unit vector n. Then, from Eq. (10.3.2), we see that the work done by C is:

θ *
∫ ⋅ dθ
W = Tn (10.3.3)
0

*
where θ is the total angle of rotation while C is being applied, and T is the torque of C.