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ilovepdf.com_split_3 - 99 Question 317 A particle of mass m...

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99 Question 3–17 A particle of mass m is attached to an inextensible massless rope of length l as shown in Fig. P3-17. The rope is attached at its other end to point A located at the top of a fixed cylinder of radius R . As the particle moves, the rope wraps itself around the cylinder and never becomes slack. Knowing that θ is the angle measured from the vertical to the point of tangency of the exposed portion of the rope with the cylinder and that gravity acts downward, determine the differential equation of motion for the particle in terms of the angle θ . You may assume in your solution that the angle θ is always positive. g m A B O R θ Figure P3-17 Solution to Question 3–17 Kinematics First, let F be a reference frame fixed to the circular track. Then, choose the following coordinate system fixed in reference frame F : Origin at O E x = Along OA E z = Out of Page E y = E z × E x Next, let A be a reference frame fixed to the exposed portion of the rope. Then, choose the following coordinate system fixed in reference frame A : Origin at O e r = Along OB e z = Out of Page e θ = e z × e r The geometry of the bases { E x , E y , E z } and { e r , e θ , e z } is shown in Fig. 3-13. Using Fig. 3-13, we have that E x = cos θ e r - sin θ e θ (3.382) E y = sin θ e r + cos θ e θ (3.383)
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100 Chapter 3. Kinetics of Particles e r e θ E x E y e z , E z θ θ Figure 3-13 Geometry of Question 3–17. Now we note that the rope has a fixed length l . Since the length of the portion of the rope wrapped around the cylinder is , the exposed portion of the rope must have length l - . Furthermore, since the exposed portion of the rope lies along the direction from B to m
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