Impulse-Momentum for Rigid BodiesLecture 31

ME 231: Dynamics

Question of the Day

A slender bar of mass 0.8 kg and length 0.4 m is falling with a velocity v = 2 m/s and angularvelocity �� = 10 rad/s.

Determine the angular momentum HO of the bar about point O.

ME 231: Dynamics

mvd�IH GO ��

Outline for Today

� Question of the day� Linear momentum for rigid bodies� Angular momentum for rigid bodies� Interconnected rigid bodies� Conservation of momentum for rigid

bodies� Answer your questions!

ME 231: Dynamics

� Question of the day

Recall: Linear Momentum for Particle Systems

ME 231: Dynamics

vG m�

GF ���

� �iim �vG ����iim vG ��

avG mm �� ��

iii mdtdm �vG ����

0� �m

iii m vG �

Linear Momentum for Rigid Bodies

� Special case of a general system of particles

ME 231: Dynamics

xx GF ���

yy GF ���

vG m�

GF ��� 21 2

1

GFG ��� dtt

t

� � � �xt

t xx GdtFG 21 2

1

��� � � � �y

t

t yy GdtFG 21 2

1

���

� � � �iiiiiG mm ���rH �� �����

Recall: Angular Momentum for Particle Systems

ME 231: Dynamics

� �� iiiG m �r�H �� ���

About the Mass Center G

GG HM ���

0 � �iiG F�H ���

� � iiiiG m r�H ��

� �iiiG m r�H ���

� � � �iiiiiG mm ��r�H �� ����

� �iiiG m ��H ���

Angular Momentum for Rigid Bodies

ME 231: Dynamics

About the Mass Center G

GG HM ���

angularlinear

vG m�

GF ���

21 2

1

GFG ��� dtt

t� � � �21 2

1G

t

t GG dt HMH ���

�H GG I�

linear

vG m�

GF ���

212

1

GFG ��� 1

dtt

t

Alternate Angular Momentum for Rigid Bodies

ME 231: Dynamics

PP HM ���

� � � �21 2

1P

t

t PP dt HMH ���

mvd�IH GP ��

About an Arbitrary Point P About a Fixed Point O

OO HM ���

� � � �21 2

1O

t

t OO dt HMH ���

�IH OO �P

ME 231: Dynamics

Recall: Systems of Interconnected Bodies

kinetic diagramfree-body diagram

aF m���

mad�IM GP �����

ME 231: Dynamics

Interconnected Rigid Bodies

combined free-body and momentum diagrams

��� ���� ba GGF

� � � � ��� ���� bOaOO HHM

� �systemO

t

t O dt HM ��� 2

1

� �systemt

tdt GF ��� 2

1

Recall: Conservation of Momentum for Systems

� If the resultant external force ��F is zero, then linear momentum is conserved

� If the resultant moment about a fixed pointO or mass center G is zero, then angular momentum is conserved

ME 231: Dynamics

� � � �21 OO HH � 21 GG � � � � �21 GG HH �

Conservation of Momentum for Rigid Bodies

� If the resultant external force ��F is zero, then linear momentum is conserved

� If the resultant moment about a fixed pointO or mass center G is zero, then angular momentum is conserved

ME 231: Dynamics

� � � �21 OO HH � 21 GG � � � � �21 GG HH �

Exactly the same as for particle systems!

Impulse-Momentum for Rigid Bodies: Exercise 1

ME 231: Dynamics

A constant horizontal force P is applied the light yoke attached to the center O of a uniform circular disk of mass m, which is initially at rest

and rolls without slipping.

Determine the velocityv of the center O in terms of t.

Impulse-Momentum for Rigid Bodies: Exercise 2

ME 231: Dynamics

A uniform circular disk has a mass of 25 kg and is mounted to a rotating bar in three different ways and ��O= 4 rad/s

Determine the angular momentum HO of the disk about point O for each case.

rad/s 8���

Impulse-Momentum for Rigid Bodies: Exercise 3

ME 231: Dynamics

Fully extended the diver’s 80-kg body has a angular velocity of 0.3 rev/s.

Determine the angular velocity �� later when the diver assumes a tuck position.

Outline for Today

� Question of the day� Linear momentum for rigid bodies� Angular momentum for rigid bodies� Interconnected rigid bodies� Conservation of momentum for rigid

bodies� Answer your questions!

ME 231: Dynamics

� Question of the day� Linear momentum for rigid bodies� Angular momentum for rigid bodies� Interconnected rigid bodies� Conservation of momentum for rigid

bodies

For Next Time…

� Exam 2.a on Friday (11/9)� Homework #10 due on Monday

(11/12)� Homework #11 due on

Wednesday(11/14)� Read Chapter 8, Sections 8.2 & 8.3

ME 231: Dynamics