Assignment 6 solution 2010

Assignment 6 solution 2010 - Analytical Mechanics CIVE 281...

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Analytical Mechanics CIVE 281 Solution to ASSIGNMENT No.6 5-158 Four pins slide in four seperate slots cut in circular plate as shown. When the plate is at rest, each pin has the velocity directed as shown and of the same constant magnitude u . If each pin maintains the same velocity relative to the plate when plate start to rotate about O with constant coun- terclockwise angular velocity ω , determine the acceleration of each point. SOLUTION: The Fixed frame is located at“ O ”, also refrence point for rotating frame is located at “ O ” which rotates with constant ω : For all four points, ω = constant , hence ˙ ω = 0 . 0 and ¨ ω = 0 . 0 Also as refrence point for both rotating and fixed frames are located at “ O ”: −→ r a/O = 0, −→ v a/O = 0, −→ a a/O = 0 a p = a a + ˙ ω × r p/A + ω × ( ω × r p/A ) + 2 ω × ( ˙ r p/A ) rf + (¨ r p/A ) rf POINT P1: a a = 0 , ˙ ω = 0 , r p/A = r ˆ i, ( ˙ r p/A ) rf = u ˆ i, r p/A ) rf = 0 , ω = ω ˆ k replacing in acceleration equation we get a p 1 = 2 ˆ i 2 ˆ j 1
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POINT P2: a a = 0 , ˙ ω = 0 , r p/A = r ˆ j, ( ˙ r p/A ) rf = u ˆ j, r p/A ) rf = 0 , ω = ω ˆ k replacing in acceleration equation we get a p 2 = 2 ˆ i 2 ˆ j POINT P4: a a = 0 , ˙ ω = 0 , r p/A = r ˆ j, ( ˙ r p/A ) rf = u ˆ i, r p/A ) rf = 0 , ω = ω ˆ k replacing in acceleration equation we get a p 4 = ( 2 + 2 ) ˆ j POINT P3: a a = 0 , ˙ ω = 0 , r p/A = r ˆ i, ( ˙ r p/A ) rf = u ˆ j, r p/A ) rf = 0 , ω = ω ˆ k r p/A ) rf is the accelaration due to curve path of motion that is toward the center: r p/A ) rf = ω 2 rf r = ( u 2 /r 2 ) r = ( u 2 /r ) replacing in acceleration equation we get a p 3 = ( 2 + 2 + u 2 r ) ˆ i 2
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5-188 A 60mm-radius disk spins at the constant rate ω 2 = 4 rad/s about the axis held by a housing attached to a horizontal rod that rotates at the constant ω 1 = 5 rad/s. For the position shown, determine (a) the angular acceleration of the disk, (b) the acceleration of point P on the rim of the disk if θ = 0 , (c) the acceleration of point P on the rim of the disk if θ = 90 .
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