EAS208_HW03_091510_Solutions

EAS208_HW03_091510_Solutions - EAS 208 (Fall 2010) Homework...

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EAS 208 (Fall 2010) Homework #3 – Solutions Problem 13–142 At the instant shown, the coordinates of the collar are A 2.3 x ft = , 1.9 y ft = . The collar is sliding on the bar from B toward C at a constant speed of 4 ft s . a) What is the velocity of the collar in terms of polar coordinates? b) Use answer to part a) to determine the angular velocity of the radial line from the origin to the collar at the instant shown. A Answer: The velocity in Cartesian coordinates is: cos60 4cos60 2 oo x ft vv s === sin 60 4sin 60 3.46 y ft s The angle θ is: 1 1.9 tan 39.56 2.3 o ⎛⎞ == ⎜⎟ ⎝⎠ 1
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The velocity in polar coordinates is: cos sin 2cos39.56 3.46sin39.56 3.75 oo rx y ft vv v s θθ =+= + = sin cos 2sin39.56 3.46cos39.56 1.39 xy ft v s θ =− + + = thus 3.75 1.39 r = + ve e ( ) ft s b) The angular velocity of the radial line from the origin to the collar A at the instant shown is: 22 1.39 0.466 2.3 1.9 ft v dd s rad vr s dt dt r ft ft =⇒ = = = + 0.466 d rad s dt ⇒= 2
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Problem 13–153 The hydraulic actuator moves the pin upwards with velocity P 2 m s = v j . Determine the velocity of the pin in terms of polar coordinates and the angular velocity of the slotted bar when . 35 o θ = Answer: The velocity of pin is: P 2 P m s = vj In polar coordinates the position and velocity vectors are given by the relations: r r = re rr r vv r r θθ =+ = + ve ee e & & 3
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Also, note that: 2 y m = + ri j 2 m y s == v jj & sin ry θ = && tan y x = 22 rx y =+ m cos d vr y r dt = ⇒= & & 35 o = Solve the above set of equations to get: sin 2sin35 1.15 o m r r s =⇒ = = & & tan tan 2 tan 35 1.4 o y y xy y x θθ =⇒= m 2 1.4 2.44 rm r = m cos 2cos35 cos 0.671 2.44 o y rad yr s r θθθ = = = & & & & Finally, in polar coordinates the velocity vector when is: 35 o = 1.15 2.44 0.671 rr r r vv r r = + = ve ee e e e & & 1.15 1.63 r + ve e ( ) m s The angular velocity of the slotted bar when is: 35 o = 0.671 d rad s dt & 4
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Problem 13–160
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EAS208_HW03_091510_Solutions - EAS 208 (Fall 2010) Homework...

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