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Relative-Motion Analysis: Velocity • Translation only Kinematics – Position rB rA rB / A – Velocity vB vA – Acceleration aB aA y A B rA rB x Relative-Motion Analysis: Velocity • Transl. & Rotation y (General Plane Motion) – Position rB rA rB / A – Velocity (time deriv) vB v A vB / A where v B / A rB / A and (ω is rotation of member about A) For our problems, we will just need to plug in for each of these variables to get vB.and often ω drA A rB/A B drA drB rA rB x dθ rB/A (new) drB/A Review of Cross Products • See Section 4.2 for full details A B Ax ˆj kˆ Ay Az Bx By Bz iˆ or To use, must define right-hand x, y, z coordinate system Example Problem If rod AB slides along the horizontal slot with a velocity of 60 ft/s, determine the angular velocity of link BC at the instant shown. (F16-11, 48 rad/s) What about the velocity of the pin at C, and the angular velocity of wheel OC at that instant? (104 ft/s up) Special Case for Velocity Solution Rolling without slip Can also have slip, in that instance direction of vA is at least known but magnitude unknown Example Problem A bowling ball is cast on the “alley” with a backspin of ω = 10 rad/s while its center O has a forward velocity of vO = 8 m/s. Determine the velocity of the contact point A in contact with the alley. (16-58, 9.20 m/s to the right) Instantaneous Center of Zero Velocity • Relate velocity of two points on right body Rolling without slip (not always) v B v A rB / A • What if choose a point A which is instantaneously stationary (i.e. vA = 0) v B rB / A ; vB r • Can we find an instant point with this property to relate to? What if we want velocity at each point on rim? (each point will instantaneously rotate about axis fixed to that point) Instantaneous Center of Zero Velocity • Does an I.C. always exist? – – – – At some instant, yes Consider curvilinear motion in particle mech. For rigid body? I.C. need not be ON the body? • To find I.C. – Identify instantaneous direction of velocity for each point – Draw perpendicular lines from each – Intersection is I.C. at that instant • To solve vPoint = ωrPoint/IC Graphic Examples To find I.C. Identify instantaneous direction of velocity for each point Draw perpendicular lines from each Intersection is I.C. at that instant To solve vPoint = ωrPoint/IC Example Problem If link CD has an angular velocity of 6 rad/s, determine the velocity of point E on link BC and the angular velocity of link AB at the instant shown. (16-89, 6 rad/s CCW, 4.76 m/s, 40.9° above – x)