Chapter 19: Planar Kinetics of a Rigid Body: Impulse and
Momentum
19.1 Linear and Angular Momentum
 The linear momentum of a rigid body of mass m is
L = m vG
(1)
where vG is the velocity of the body’s mass center.
 The magnitude of the angular momentum of a rigid body about the body’s center of mass is
HG = IG ω
(2)
where
IG = moment of inertia of the body about an axis perpendicular to the plane of
motion and passing through the mass center
ω = angular velocity of the body
Note: Angular momentum is a vector quantity. It’s direction is defined by ω.
 The angular momentum of the body computed about point P (shown below) is
H p   y m vG x  x m vG  y  I G 
(3)
That is, the angular momentum computed about point
P is equivalent to the moment of the linear
momentum mvG or its components m(vG)x and m(vG)y
about P plus the angular momentum IGω.
Translation
 When the rigid body is subjected to translation only, ω = 0 for the body. Hence the linear
momentum and the angular momentum computed about G become
L  m vG
HG  0
(4)
 If the angular momentum is computed about point A (shown in Fig. 19-2a), then HA =
(d)(mvG), where d is the moment arm.
Rotation about a Fixed Axis
 When the rigid body is rotating about a fixed axis passing through point O (see Fig. 19-2b),
the linear momentum and the angular momentum computed about G are
L  m vG
(5)
HG  IG 
 If the angular momentum is computed about point O, then
H O  I G   rG m vG 

(6)

 I G  m rG2 
Therefore,
H O  I O
(7)
General Plane Motion
 When the rigid body is subjected to general plane motion, the linear momentum and the
angular momentum computed about G become
L  m vG
HG  IG 
(8)
 If the angular momentum is computed about point A (shown in Fig. 19-2c), then HA = IGω +
(d)(mvG), where d is the moment arm.
19.2 Principle of Impulse and Momentum
 Like the case of particle motion (Chapter 15), the principle of impulse and momentum for a
rigid body is used for a direct solution to problems involving force, velocity, and time.
 The principle of linear impulse and momentum can be expressed as follows

t2
t1
F dt  mv G 2  mv G 1
(9)
That is, the sum of all the impulses created by the external force system which acts on the
body during the time interval t1 to t2 is equal to the change in the linear momentum of the
body during the time interval.
 The principle of angular impulse and momentum for a body under general plane motion can
be expressed as

t2
t1
M G dt  I G 2  I G 1
(10)
 Similarly, for rotation about a fixed axis passing through point O, we have

t2
t1
M O dt  I O  2  I O 1
(11)
That is, the sum of the angular impulses acting on the body during the time interval t1 to
t2 is equal to the change in the body’s angular momentum during this time interval.
 Therefore, the following three scalar equations may be written to describe the planar motion
of the body.
mvG x 1 

t2
mvG y 1 

t2
I G 1 
t1
t1

t2
t1
Fx dt  mvG x 2
Fy dt  mvG y 2
(12)
M G dt  I G 2
19.3 Conservation of Momentum
Conservation of Linear Momentum
 If the sum of all the linear impulses acting on a system of connected rigid bodies is zero, the
linear momentum of the system is conserved. Consequently, the first two equations of
equation 19.15 of the textbook reduce to the form
 syst . linear 
 syst . linear 
 
   

 momentum  1  momentum  2
 Equation (13) is referred to as the conservation of linear momentum.
(13)
 Equation (13) can also be applied in a specified direction for which the linear impulses are
small or nonimpulsive. [See details of nonimpulsive forces in Chapter 15.]
Conservation of Angular Momentum
 The angular momentum of a system of connected rigid bodies is conserved about the
system’s center of mass G, or a fixed point O, when the sum of all angular impulses created
by the external forces acting on the system is zero or appreciably small (nonimpulsive) when
computed about these points. The third of equations 19.15 then becomes
 syst . angular 
 syst . angular 
 
   

 momentum  O1  momentum  O 2
(14)
 Equation (14) is referred to as the conservation of angular momentum.
19.4 Eccentric Impact
 Eccentric impact occurs when the line connecting the mass centers of the two bodies does not
coincide with the line of impact. This often occurs when one or both of the bodies are
constrained to rotate about a fixed axis.
 To obtain the velocities of the two bodies along the line of impact after collision, two
equations are applied. These are
1. Conservation of angular momentum of the system of bodies
2. The definition of the coefficient of restitution. For the two bodies in Fig. 19-11, the
coefficient of restitution can be written as
e
v B 2  v A 2
v A 1  v B 1
(15)
[Application of this principle will be illustrated in class with the discussion of problem 19-51.]