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p0709

p0709 - r is its position vector Now we know that ˙ Ω =...

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314 CHAPTER 7. RIGID-BODY KINEMATICS 7.9 Chapter 7, Problem 9 Problem: Rod AB is rotating in the counterclockwise direction with constant angular velocity ω . Collar P slides without friction as shown with constant relative speed u . If r = 1 2 f when θ = 0 o and the collar reaches Point B where r = f when θ = 90 o , what is Collar P’s acceleration vector at the instant it reaches Point B? Express your answer as a function of ω and f . Solution: We know that in a coordinate frame rotating at angular velocity = ω k about Point A, the collar is not accelerating, i.e., a I = 0 where a I is the acceleration seen by a rotating observer. In general, the collar’s absolute acceleration is a = a I + ˙ × r + 2 × v I + × ( × r ) where v I is the collar’s velocity as seen by the rotating observer and
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Unformatted text preview: r is its position vector. Now, we know that ˙ Ω = , v I = u e r , r = f e r where e r is a unit vector in the radial direction. Hence, a = 2 ω k × u e r + ω k × ( ω k × f e r ) = 2 ω u e θ + ω 2 f k × e θ = 2 ω u e θ − ω 2 f e r where e θ is a unit vector in the circumferential direction. Finally, we can express u as a function of ω and f from the given information. That is, we know that u = dr dt = dr d θ d θ dt = w f − 1 2 f π / 2 − W ω = ωf π Therefore, the collar’s acceleration is a = − ω 2 f e r + 2 π ω 2 f e θ...
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