Relative Velocity
Lecture 11
ME 231: Dynamics
Question of the Day
ME 231: Dynamics
The velocity of the cart is 4 ft/s to the right.
Determine the angular speed N (in rpm’s) of the wheel so that point A on the top of the rim has a velocity equal to zero.
Outline for Today
� Question of the day� Relative velocity due to rotation
� Interpretation of vA = vB + vA/B� Solution of relative-velocity equation� Answer your questions!
ME 231: Dynamics
� Question of the day
Recall: Relative Motion
ME 231: Dynamics
� Absolute position of B is defined in an inertial coordinate system X-Y
� Attach a set of translating (non-rotating) axes x-y to particle B and define the position of A
� Define position of “A relative to B” (“A/B”) in x-y
?�Ar
?�Av
?�Aa
BABA /rrr ��
BABAA /rrrv ��� ���
BABAAA /rrrva ������� ����
Relative Velocity Due to Rotation
� Movement in two parts
� First, body translates��rB to parallel position
� Second, body rotates about B’ through angle ��
ME 231: Dynamics
From translating (non-rotating) axes x’-y’ attached to point B’, the remaining motion is a simple rotation about B’ giving ��rA/B
BABA /rrr �����
Relative Velocity Due to Rotation
ME 231: Dynamics
BABA /rrr �����
Dividing by ��tBABA /vvv ��
Relative velocity as a scalar
tv BA
tBA ��
���
/
0/ limr
tr
t ��
���
�0
lim
…as a vector
�r�
r�v ��BA /
Interpretation of Relative-Velocity Equation
ME 231: Dynamics
BABA /vvv ��
Translational portion
Rotationalportion
r�v ��BA /Bv
Bv
BA /v
Av
90º
Outline for Today
� Question of the day� Relative velocity due to rotation
� Interpretation of vA = vB + vA/B� Solution of relative-velocity equation� Answer your questions!
ME 231: Dynamics
� Question of the day� Relative velocity due to rotation
� Interpretation of vA = vB + vA/B
Solution of Relative-Velocity Equation: Case #1
ME 231: Dynamics
Scalar-Geometric
OAOA /vvv ��
Ov
OA /vAv
vA /O � rO� � rOvOr
��
��� 2 m/s
3 m/s
2 m/s 60º
vA2 � 22 � 32 � 2 2� � 3� �cos120�vA � 4.36 m/s
�10 rad/s
r0
vA2 � vA /O
2 � vO2 � 2 vA /O� � vO� �cos120�
vA � vO � vA /O � vO ��� � r0
Solution of Relative-Velocity Equation: Case #2
ME 231: Dynamics
Vector
OAOA /vvv ��
3 m/s30º Ov
OA /vAv
m/s 7.147.1301.017.010003 jiijikji
iv ������
���A
m/s 36.4�Av
�10 rad/s
r0
Relative Velocity: Exercise
A mechanic “walks” a two-tire unit that rolls without slipping at constant speed v. The radius of both tires is r.
ME 231: Dynamics
Determine the velocitiesof points A, B, C, and D.
Outline for Today
� Question of the day� Relative velocity due to rotation
� Interpretation of vA = vB + vA/B� Solution of relative-velocity equation� Answer your questions!
ME 231: Dynamics
� Question of the day� Relative velocity due to rotation
� Interpretation of vA = vB + vA/B� Solution of relative-velocity equation
For Next Time…
� Complete Homework #4 due on Thursday (9/20)
� Read Chapter 6, Section 6.2
ME 231: Dynamics