As shown in Figure CQ5.22, student A, a 55-kg girl, sits on one chair with metal
ID: 1266314 • Letter: A
Question
As shown in Figure CQ5.22, student A, a 55-kg girl, sits on one chair with metal runners, at rest on a classroom floor. Student B, an 80-kg boy, sits on an identical chair. Both students keep their feet off the floor. A rope runs from student A's hands around a light pulley and then over her shoulder to the hands of a teacher standing on the floor behind her. The low-friction axle of the pulley is attached to a second rope held by student B. All ropes run parallel to the chair runners, (a) If student A pulls on her end of the rope, will her chair or will B's chair slide on the floor? Explain why. (b) If instead the teacher pulls on his rope end, which chair slides? Why this one? (c) If student B pulls on his rope, which chair slides? Why? Now the teacher ties his end of the rope to student A's chair. Student A pulls on the end of the rope in her hands. Which chair slides and why?Explanation / Answer
Analysis:
Here are some Principles that apply to this problem; most can be derived from Reference 1:
Principles P3-P5 apply until one of the chairs starts moving.
P1: The tension T1 in the first rope is the same at all points along the rope because it is not stretching and the pulley has "low" (negligible) friction.
P2: Whoever pulls on a rope creates a tension in both ropes.
P3: The tension in the second rope T2 is twice the tension T1 in the first rope.
P4: The sliding of the two chairs on the floor is governed by the same coefficient of static friction ?s and the same gravitational acceleration g, so the force due to static friction Ff of each chair is equal to a constant K ( = ?s * g) X mass of the person sitting in the chair (assuming negligible mass of the chair).
P5: The direction of each chair's frictional force Ff is opposite and equal to the tension of the rope(s) attached to that chair.
Answers:
(a) If student A pulls on her end of the the rope, will her chair or will B's chair slide on the floor?
In other words, as A increases the tension T1, which chair will slide first?
Because of P3 (i.e., T2 = 2 * T1) and P4, Student A only has to exert a T1 of slightly more than 80K / 2 = 40K to move Student B's chair. Student A will not move until T1 > 55K, so B's chair moves first.
(b) If instead the teacher pulls on his rope end, which chair slides?
The teacher's pulling on his rope creates tensions T1 and T2, just as Student A did in Question (a). So the answer is the same as in (a)--Student B's chair moves first.
(c) If student B pulls on his rope, which chair slides? Why?
Student B pulling on his rope creates tensions T1 and T2, just as Student A did in Question (a). Because T2 = 2 * T1, he will have to pull twice as hard as Student A did, but the static tension relationships are still the same. So the answer is the same as in (a)--Student B's chair moves first.
(d) Now the teacher ties his end of the rope to student A's chair. Student A pulls on the end of the rope in her hands. Which chair slides and why?
Here the situation is slightly different: With the teacher gone, the force pulling Student A's chair is now 2 * T1 (because there are two rope connections to her chair, each with tension T1). This is the same force pulling Student B's chair.
Since the same magnitude of force is pulling on both chairs, the chair with the smaller Ff will move first; and that's Student A's chair, with FfA = 55K N, which is less than FfB = 80K N.
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