Relative Velocity

Lecture 11

ME 231: Dynamics Question of the Day

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.

ME 231: Dynamics Outline for Today

Question ofof the dayday Relative velocity due to rotation

Interpretation of vA = vB + vA/B Solution of relative-velocity equation Answer your questions!

ME 231: Dynamics Recall: Relative Motion

rA  r?B  rA/ B

   v A  r?A  rB  rA/ B

    a A  ?v A  rA  rB  rA/ B

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 ME 231: Dynamics Relative Velocity Due to Rotation

Movement in two parts First, body translates rB to parallel position rA  rB  rA/ B Second, body rotates about From translating (non-rotating) B’ through axes x’-y’ attached to point B’, angle  the remaining motion is a simple rotation about B’ giving rA/B ME 231: Dynamics Relative Velocity Due to Rotation

Relative velocity as a scalar

rA/ B vA/ B  lim t 0 t r  lim t 0 t r  r  r  r A B A/ B

…as a vector v A  v B  v A/ B Dividing by t

v A/ B  r

ME 231: Dynamics Interpretation of Relative-Velocity Equation

Translational Rotational portion portion 90º

v  r v B A/ B

v A/ B

v A  v B  v A/ B v A v B

ME 231: Dynamics Outline for Today

Question ofof the dayday RelativeRelative vevelocitylocity ddueue to rotationrotation

InterpretationInterpretation ofof vA = vB + vA/BA/B Solution of relative-velocity equation Answer your questions!

ME 231: Dynamics Solution of Relative-Velocity Equation: Case #1 C v S Scalar-Geometric v  r   r D O T  2 m/s A /O O OE r U 2 2 2 vA  vA /O  vO  2vA /O vO cos120 2 2 2 vA  2  3  223 cos120 v  4.36 m/s A v A/O v A 2 m/s 60º r0 vO 3 m/s v  v  v 10 rad/s A O A/O

ME 231: Dynamics Solution of Relative-Velocity Equation: Case #2 Vector vA  vO  vA /O  vO   r0 i j k v A  3i  0 0 10  3i 1.7j i  4i 1.7j m/s  0.17 0.1 0 v v  4.36 m/s A/O A v A

r0 vO 30º 3 m/s v  v  v 10 rad/s A O A/O

ME 231: Dynamics 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.

Determine the velocities of points A, B, C, and D.

ME 231: Dynamics Outline for Today

Question ofof the dayday RelativeRelative vevelocitylocity ddueue to rotationrotation

InterpretationInterpretation ofof vA = vB + vA/BA/B SolutionSolution of relative-velocityrelative-velocity eequationquation Answer your questions!

ME 231: Dynamics For Next Time…

Complete Homework #4 due on Thursday (9/20) Read Chapter 6, Section 6.2

ME 231: Dynamics