Jack Sparrow, Will Turner, and James Norrington each want Davy Jones' heart for their own ends....

 

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Jack Sparrow, Will Turner, and James Norrington each want Davy Jones' heart for their own ends. The dead man's heart is hidden in a chest buried on Isla Cruces, but one needs the key to unlock the chest and procure the heart. While fighting each other for possession of the key, Will and Norrington have dislodged a large water wheel from an old, dilapidated building, and they continue to fight on top of it as it rolls along the ground. Meanwhile, through a series of improbable events, the key has become hooked on a nail on the interior of the wheel, and Jack chases after the wheel while the other two duel on its top5. Eventually, the three adversaries all enter into the interior of the wheel and fight over the key. (See Figure 4 below.) Actual problem statement: Determine the equation of motion of the system, where each person is modeled as a particle and the wheel is modeled as a point mass (a particle). Each person is holding on for dear life, so they are all fixed to the wheel in their re- spective positions. Consequently, they each have reaction forces that do not necessarily point towards the center of the wheel. The wheel has radius R, and the angle, 0, is measured relative to the plane of the ground. Assume the wheel rolls on the ground without slip; kinetically, this kinematic constraint is enforced via a "rolling force" act- ing horizontally on the wheel; this rolling force is effectively static friction-treat it as an unknown reaction force. Note: The three men exert no forces on each other, so we shall allow you to only draw five free-body diagrams: one for each combatant, one for the wheel, and one for the entire system. Extra Note: We are modeling the wheel as a point mass in this problem, and we will say its position is at the center of the wheel, i.e., the positions, velocities, and accelerations of Will Turner and the wheel are the same, so you do not need to show that work twice. Wheel of mass m4 (point mass located at the center) Will Turner 90 at the center mi Formerly Commodore Norrington R (m2) m3 Captain Jack Sparrow Figure 4: Water wheel in which Jack, Will, and Norrington are fighting for problem 3. Jack Sparrow, Will Turner, and James Norrington each want Davy Jones' heart for their own ends. The dead man's heart is hidden in a chest buried on Isla Cruces, but one needs the key to unlock the chest and procure the heart. While fighting each other for possession of the key, Will and Norrington have dislodged a large water wheel from an old, dilapidated building, and they continue to fight on top of it as it rolls along the ground. Meanwhile, through a series of improbable events, the key has become hooked on a nail on the interior of the wheel, and Jack chases after the wheel while the other two duel on its top5. Eventually, the three adversaries all enter into the interior of the wheel and fight over the key. (See Figure 4 below.) Actual problem statement: Determine the equation of motion of the system, where each person is modeled as a particle and the wheel is modeled as a point mass (a particle). Each person is holding on for dear life, so they are all fixed to the wheel in their re- spective positions. Consequently, they each have reaction forces that do not necessarily point towards the center of the wheel. The wheel has radius R, and the angle, 0, is measured relative to the plane of the ground. Assume the wheel rolls on the ground without slip; kinetically, this kinematic constraint is enforced via a "rolling force" act- ing horizontally on the wheel; this rolling force is effectively static friction-treat it as an unknown reaction force. Note: The three men exert no forces on each other, so we shall allow you to only draw five free-body diagrams: one for each combatant, one for the wheel, and one for the entire system. Extra Note: We are modeling the wheel as a point mass in this problem, and we will say its position is at the center of the wheel, i.e., the positions, velocities, and accelerations of Will Turner and the wheel are the same, so you do not need to show that work twice. Wheel of mass m4 (point mass located at the center) Will Turner 90 at the center mi Formerly Commodore Norrington R (m2) m3 Captain Jack Sparrow Figure 4: Water wheel in which Jack, Will, and Norrington are fighting for problem 3.