Page 312 - Dynamics of Mechanical Systems

0593_C09_fm Page 293 Monday, May 6, 2002 2:50 PM

Principles of Impulse and Momentum 293

From d’Alembert’s principle, the entire force system of a free-body diagram is a zero
system. Thus, the moment of the force system about any and all points is zero. Hence,
consider setting moments about O, Q, and G equal to zero: for O we have:

P ×+ F + M + T = 0 (9.6.25)
F P ×
*
*
G G G
*
By substituting for F and T from Eq. (9.6.24) we have:
*
M = P × m a + I B G ⋅ +αωω × I ( B G ⋅ ) (9.6.26)
α
ωω
G
O G
where M is defined as M + P × F and is identified as the moment of the applied forces
G
O
about O. By comparing Eqs. (9.6.21) and (9.6.26) we have:
M = R d A dt (9.6.27)
O BO
Next, by setting moments about Q equal to zero we obtain:

QG F M + QG F + T = 0 (9.6.28)
×
×+
*
*
G
Then, by substituting from Eq. (9.6.24), we have:
×
α
ωω
M = QG m a + I B G ⋅ +αωω × I ( B G ⋅ ) (9.6.29)
G
Q
where M is defined as QG × F + M and is identified as the moment of the applied forces
Q
G
about Q. By comparing Eqs. (9.6.22) and (9.6.29), we have:
M = R d A dt + v × m v G (9.6.30)
Q
Q BQ
Finally, by setting moments about G equal to zero we have:
M + T = 0 (9.6.31)
*
G
Hence, by substituting from Eq. (9.6.8) we have:

ωω
M + I BG ⋅ +αωω × I ( BG ⋅ ) (9.6.32)
α
G
By comparing Eqs. (9.6.23) and (9.6.32), we obtain:

M = R d A dt (9.6.33)
G BG

By inspecting Eqs. (9.6.27), (9.6.30), and (9.6.33), we see that if the object point is either
fixed in the inertia frame R or is the mass center of the body, then the moment of the
applied forces about the object point is equal to the derivative of the angular momentum
about the object point.