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