planar kinetics of a rigid body impulse and momentum · 2010-09-04 · planar kinetics of a rigid...
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Planar Kinetics of a Rigid BodyImpulse and Momentum
Momentum
• Linear momentum of any particle ith:
• Linear momentum of a rigid body:
04/09/53 ME212 ดร. พภัิทร 2
Linear Momentum
ii vL im
G
ii
vL
vLL
m
mi
Linear and Angular Momentum
• Consider angular momentum about P
04/09/53 ME212 ดร. พภัิทร 3
Angular Momentum
iiiP m vrH
rvvvv PPiPi /
Linear and Angular Momentum
• Using Cartesian vectors
• Let mi → dm and integrating over the entire mass m of the body
04/09/53 ME212 ดร. พภัิทร 4
2)()(
)()(
rmvxmvymH
yxvvyxmH
iyPixPiiP
yPxPiiP
jikjijik
m
yPm
xPm
P dmrvdmxvdmyH 2)()(
Integrals locating G of the body Moment of Inertia @ P
Linear and Angular Momentum
• Thus
04/09/53 ME212 ดร. พภัิทร 5
PyPxPP IvmxvmyH )()(
GyGxGP IvmxvmyH )()(
PGPG /rvv
Parallel-axis theorem
GG IH 0,0, yxGP
Linear and Angular Momentum
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Translation
Linear Momentum L =
Angular Momentum HG =
Angular Momentum HA =
Linear and Angular Momentum
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Rotation about a Fixed Axis
Linear Momentum L =
Angular Momentum HG =
Angular Momentum HO =
Linear and Angular Momentum
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General Plane Motion
Linear Momentum L =
Angular Momentum HG =
Angular Momentum HA =
Principle of Impulse and Momentum
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)( Gmdt
dvF
12
2
1)()( GG
t
tmmdt vvF
Linear Momentum
Angular Momentum
)( Gmdt
dvrFr )( GG I
dt
dM
12
2
1
G
t
tGG IIdtM
Principle of Impulse and Momentum
04/09/53 ME212 ดร. พภัิทร 10
2
2
11
2
2
11
2
2
11
)()(
)()(
G
t
tGG
Gy
t
tyGy
Gx
t
txGx
IdtMI
vmdtFvm
vmdtFvm
+ =
Principle of Impulse and Momentum
04/09/53 ME212 ดร. พภัิทร 11
2)21(1
2)21(1
2)21(1
...
...
...
OOO
yyy
xxx
momentum
angularsyst
impulse
angularsyst
momentum
angularsyst
momentum
linearsyst
impulse
linearsyst
momentum
linearsyst
momentum
linearsyst
impulse
linearsyst
momentum
linearsyst
System of bodies
Example 1
The 100-N disk is assumed to be uniform and is pin supported at its center. If it is acted upon by a constant couple moment of 6 N.m and a force of 50 N which is applied to a cord wrapped around its periphery, determine the angular
velocity of the disk two seconds
after starting from rest.
Also, what are the force components
of reaction at the pin?
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Problem 19-12
The spool has a mass of 30 kg and a radius of gyration kO = 0.25m. Block A has a mass of 25 kg, and block B has a mass of 10 kg. If they are released from rest. Determine the time required for
block A to attain a speed of 2 m/s.
Neglect the mass of the rope.
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Conservation of Momentum
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21
21
21
..
..
..
OO
yy
xx
momentum
angularsyst
momentum
angularsyst
momentum
linearsyst
momentum
linearsyst
momentum
linearsyst
momentum
linearsyst
21
21
21
)()(
)()(
OO
yGyG
xGxG
HH
mvmv
mvmv
Problem 19-38
The rod has a length L and mass m. A smooth collar having a negligible size and one-fouth the mass of the rod is placed on the rod at its midpoint. If the rod is freely rotating at ω about its end and the collar is
released, determine the rod’s angular velocity just before the collar flies off the rod. Also, what is the speed of the collar as it leaves the rod?
04/09/53 ME212 ดร. พภัิทร 15
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