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polar_kinematics

# polar_kinematics - 3 2 2 = 12 meters ˙ r = 3 2 ˙ = 3 2 2...

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Using a Polar Description in Kinematics The position, velocity and acceleration of a point P using a polar kinematical description are given by: r P = re r v P = ˙ r e r + r ˙ " e " a P = ˙ ˙ r " r ˙ # 2 ( ) e r + r ˙ ˙ # + r ˙ # ( ) e # where e r and e " are the polar unit vectors, as shown below. Note that e r points from the fixed origin O toward P. e " is perpendicular to e r with the same positive sense as the defined angle θ . Example Point P travels on a path described by r = 3 " 2 (r is measured in meters and θ in radians). If ˙ " = 3 rad / sec = constan t what is the speed and magnitude of acceleration of P when " = 2radians . Solution
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Unformatted text preview: 3 ( ) 2 ( ) 2 = 12 meters ˙ r = 3 2 ( ) ˙ = 3 ( ) 2 ( ) 2 ( ) 3 ( ) = 36 m / sec ˙ ˙ r = 6 ˙ 2 + ˙ ˙ ( ) = 6 3 ( ) 2 + 2 ( ) ( ) [ ] = 54 m / sec 2 Therefore, v P = ˙ r e r + r ˙ e = 36e r + 12 ( ) 3 ( ) e = 36e r + 36e ( ) m / sec # v P = 36 ( ) 2 + 36 ( ) 2 = 36 2 m / sec a P = ˙ ˙ r " r ˙ 2 ( ) e r + r ˙ ˙ + 2˙ r ˙ ( ) e = 54 " 12 ( ) 3 ( ) 2 [ ] e r + 12 ( ) ( )+ 2 36 ( ) 3 ( ) [ ] e = " 54e r + 216e ( ) m / sec 2 \$ a P = 54 ( ) 2 + 216 ( ) 2 m / sec 2 r θ P O e r e θ...
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