Planar Kinetics of a Rigid BodyImpulse and Momentum

Momentum

• Linear momentum of any particle ith:

• Linear momentum of a rigid body:

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Linear Momentum

ii vL im

G

ii

vL

vLL

m

mi

Linear and Angular Momentum

• Consider angular momentum about P

28/08/54 ME212 ดร. พิภัทร 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

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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

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PyPxPP IvmxvmyH )()(

GyGxGP IvmxvmyH )()(

PGPG /rvv

Parallel-axis theorem

GG IH 0,0, yxGP

Linear and Angular Momentum

• For a planar motion on x-y plane

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GyGxGP IvmxvmyH )()(

kImH GPˆ GG/P vr

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

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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

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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?

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