MEAM 211MEAM 211
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MEAM 211: Impulse and Momentum - Angular Momentum and Inertia Response Diagrams, Study notes of Mechanical Engineering
A portion of a mechanical engineering course notes focusing on impulse and momentum, linear and angular, and the kinetics of planar rigid bodies. It includes formulas for linear and angular impulse and momentum, force and moment balance, and angular momentum about different points. Examples are provided to determine angular velocities after balls are released from a spinning rod or when a sphere hits a rod.
Typology: Study notes
Pre 2010
1 / 10
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MEAM 211MEAM 211 2
P O
Q
P O
Q o 30 o 45^ i j
MEAM 211MEAM 211
P
O^
C
MEAM 211MEAM 211
Linear/Angular Impulse and Momentum ^ Force balance ^ Moment balanceor, for a
fixed point
dGm dt ,^ O
N i i^
v
F
F^
=^ ∑ =^1
G
d^ H^ G M = dt
O
d^ H^ O M = dt
^ Linear momentum
(^ )^
(^ )^1 2 2 21 1
t
mt
mdt
LI^
G G t t
v
v
F^
=^ −∫ ^ Angular momentum
(^ )^
(^ )^1 2 2 21 1
t
t
dt
AI^
G G t G t G^
H
H
M^
=^ −∫
(^ )^
(^ )^1 2 2 21 1
t
t
dt
AI^
O O t O t O^
H
H
M^
=^ −∫ or, for a
fixed point
,^ O
MEAM 211MEAM 211
Angular momentum about O and G
v G O ω
Angular momentum about
O
Angular momentum about
G
Angular momentum ofparticle of mass
m^ at
G
MEAM 211MEAM 211 Example 1
Two solid spheres (radius
r^ = 3 in.,
mass^ m
,^ W^ = 2 lb) are mounted on a spinning horizontal rod (
R ) with
angular velocity
ω^ = 6 rad/sec as shown. The mass moment of inertiaof the rod is
I^. The balls are held R together by a string which issuddenly cut.
2 sft
lb0.
= IR
Key Observation:• None of the external forces produce amoment about the
y^ axis, the angular Determine the angular velocity of therod after the balls have moved tomomentum is conserved.• The angular momentum consists of theangular momenta of the spheres and theangular momentum of the rod
A’
and^ B’.
MEAM 211MEAM 211 Example 2
A 2-kg sphere with an initial velocityof 5 m/s strikes the lower end of an 8-kg rod
AB. The rod is hinged at
A^ and
initially at rest. The coefficient ofrestitution between the rod and sphereis 0.8.Determine the angular velocity of therod and the velocity of the sphereimmediately after impact.
MEAM 211MEAM 211
FBD/IRD for the system
Rx
Ry
mg^
I^ α^ G