Angular Impulse
and Momentum
Lecture 28
ME 231: Dynamics
Question of the Day
Determine the
magnitude HO of the
angular momentum
of the 2-kg sphere
about point O.
Hint: HO is the
moment of linear
momentum about
point O.
HO
ME 231: Dynamics
r u mv
Outline for Today
Question of the day
Angular momentum
Rate of change of angular momentum
Angular impulse-momentum principle
Plane-motion applications
Conservation of angular momentum
Answer your questions!
Exam 2 Survey
ME 231: Dynamics
Angular Momentum
HO
r u mv
Particle of mass m
is located by
position vector r
Velocity v and linear
momentum G = mv
are tangent to its path
The moment of the linear momentum vector
mv about point O is the angular momentum
HO of P about O
Perpendicular to plane A defined by r and v
ME 231: Dynamics
Angular Momentum
HO
i
x
mv x
H O x
H O y
H O z
j
y
mv y
HO
k
z
mv z
m (v z y v y z )
m(v x z v z x )
m(v y x v x y )
ME 231: Dynamics
r u mv
Rate of Change of
Angular Momentum
HO
r u mv ¦ M O
H
O
Differentiate HO
with respect to
time
H
O
H
O
H
O
r u mv r u mv
0
v u mv r u mv
r u mv
Resultant 6MO of all moments
of forces on m is the vector
cross product of r and 6F
6M O
r u 6F r u mv
ME 231: Dynamics
6M O x
6M O y
6M O z
H H H O x
O y
O z
Angular ImpulseMomentum Principle
³
t2
t1
6M O dt
H O 1 ³t
t2
1
³
t2
t1
dt
H
O
6M O dt
H O 2
HO
r u mv ¦ M O
H
O
Integrate to describe
the effect of the angular
impulse 6MO*t on
angular momentum HO
of m about O over a
finite period of time
m(v z y v y z )1 ³ 6M O x dt
t2
t1
t2
m(v x z v z x)1 ³ 6M O y dt
t1
t2
m(v y x v x y )1 ³ 6M O z dt
t1
ME 231: Dynamics
m(v z y v y z ) 2
m(v x z v z x ) 2
m(v y x v x y ) 2
Plane-Motion
Applications
H O 1 ³t
t2
1
6M O dt
HO
r u mv ¦ M O
H
O
H O 2
Moments taken about
a single axis normal
to the plane of
motion
Angular momentum
may change
magnitude and
sense, but the
direction is constant
t2
mv1d1 ³ 6Fr sin T dt
ME 231: Dynamics
t1
mv2 d 2
Conservation of
Angular Momentum
HO
r u mv ¦ M O
'H O
H O 1 ³t
t2
1
6M O dt
H O 2
H
O
0
or
H O 1 H O 2
If the resultant moment about a fixed point
O is zero, then angular momentum remains
constant, or is said to be conserved
Angular momentum may be conserved in
one coordinate (e.g., x), but not
necessarily in others (e.g., y or z)
ME 231: Dynamics
Angular Impulse-Momentum: Exercise
The particle of mass m is launched from point
O with a horizontal velocity u at time t = 0.
Determine its angular
momentum HO relative
to point O as a function
of time.
ME 231: Dynamics
Angular Impulse-Momentum: Another Exercise
The assembly starts
from rest and reaches
an angular speed of
150 rev/min under
the action of a 20-N
force T applied to the
string for t seconds.
Neglect friction and all
masses except those of
the four 3-kg spheres.
Determine t.
ME 231: Dynamics
Angular Impulse-Momentum:
Yet Another Exercise
A 0.1-kg
particle with
a velocity of 2
m/s in the
x-direction at A is
guided by a curved rail. The radius of curvature
of the rail at B is 500 mm.
Determine the time rate of change of the
angular momentum HO about the z-axis
through O at both A and B.
ME 231: Dynamics
Outline for Today
Question of the day
Angular momentum
Rate of change of angular momentum
Angular impulse-momentum principle
Plane-motion applications
Conservation of angular momentum
Answer your questions!
Exam 2 Survey
ME 231: Dynamics
For Next Time…
Begin Homework #10 due on
Wednesday (11/7)
Read Chapter 5, Section 5.3
Read Chapter 8, Section 8.2
ME 231: Dynamics