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0593_C09_fm Page 285 Monday, May 6, 2002 2:50 PM

Principles of Impulse and Momentum 285

Finally, by substituting from Eqs. (9.4.8) and (9.4.13) into Eq. (9.4.7) we obtain the
addition theorem for angular momentum for a rigid body:

A = A + A (9.4.14)
BQ G Q B G

9.5 Principle of Linear Impulse and Momentum
Consider again a particle P with mass m moving in an inertial reference frame R as in
Figure 9.5.1. Let P be acted upon by a force F as shown. Then, by Newton’s law (see Eq.
(8.3.1)), F is related to the acceleration a of P in R by the expression:

F = m a (9.5.1)
Recalling that the acceleration is the derivative of the velocity we can use the definition
of linear impulse of Eqs. (9.2.1) and (9.2.3) to integrate Eq. (9.5.1). That is,

t 2 t 2 t 2
∫
∫
)
I = ∫ Fdt = m adt = m v (d dt dt
(9.5.2)
t 1 t 1 t 1
v() − mt
t
t
= mt v() = L() − L()
2 1 2 1
or
I =∆ L (9.5.3)
where I represents the impulse applied between t and t .
1
2
Equation (9.5.3) states that the linear impulse is equal to the change in linear momentum.
This verbal statement is often called the principle of linear momentum.
This principle is readily extended to systems of particles and to rigid bodies. Consider
first the system S of N particles P with masses m (i = 1,…, N) and moving in an inertial
i
i

FIGURE 9.5.1
A particle P with mass m moving in
an inertial reference frame.