7.6 Polar Coordinates (r − θ)
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Figure 7.52: Example 7.27 ([3], pp. 80) As shown in fig. 7.53, we may expressed both the velocity and the acceleration using r-θ coordinate system. Therefore, the rate v˙ may be determined from the tangential acceleration: at = −a cos 60 = v˙ = −0.778 m/s2 One may conjecture that the rate r¨ may be determined using the expression of the radial acceleration. However, the rate θ˙ in the formula is not known yet. It may be found from the velocity relationship: ˙ θ˙ = 2.7 × 10−4 rad/s vθ = v cos 30 = r θ, Substituting its value into the following relation, r¨ may now be solved. ar = a = −1.556 = r¨ − r θ˙2 , r¨ = −0.388 m/s2 Example 7.28 ([4], Prob. 2/163) Pin A moves in a circle of 90 mm radius as crank AC revolves at the constant rate β˙ = 60 rad/s. The slotted link rotates about point O as the rod attached to A moves in and out of the slot. For the ˙ and θ. ¨ position β = 30◦ , determine r, ˙ r¨, θ, Solution: Motion of point A may be determined naturally using n-t coordinate system. Alternatively, the r-θ coordinate system may be used. At the current posture, two frames are oriented relative to each other as shown in fig. 7.55. Accordingly, the velocity lies along the t-axis and, due to the constant Chulalongkorn University
Phongsaen PITAKWATCHARA