Cuesta College, San Luis Obispo, CA
Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 8.19
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A climber is approximated as a uniform beam, with a pivot point at the feet, and a rope attached to the center. The climber is about to rappel back down from a cliff edge, leaning back with the rope initially horizontal. A little later, the climber is lower, leaning back with the same angle, but the rope is no longer horizontal. Discuss why there is less tension in the rope for the latter case. (Ignore stretching in the rope.) Explain your reasoning using diagram(s) with locations of forces and lever arms, the properties of torques, and Newton's laws.
Solution and grading rubric:
Correct. Understands that (1) the counterclockwise torque due to weight is the same in either case, such that from the static equilibrium condition (Newton's first law for rotations), the clockwise torque due to the rope's tension force must be the same in either case, and (2) the larger lever arm r⊥ for the second case will mean a smaller tension force.
As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. Does not explicitly explain in (1) why the clockwise torques due to the rope tension forces are the same.
Nearly correct, but argument has conceptual errors, or is incomplete. Discusses that r⊥ would be larger for the second case, but does not explicitly explain this on a diagram and/or with trigonometry.
Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some attempt at manipulating the relationship between torque, force, and lever arm.
Implementation/application of ideas, but credit given for effort rather than merit. Approach does not substantively use relationship between torque, force, and lever arm.
Irrelevant discussion/effectively blank.
Sections 70854, 70855, 73320
Exam code: finaln0M3
p: 6 students
r: 14 students
t: 19 students
v: 17 students
x: 2 students
y: 1 student
z: 2 students
A sample "p" response (from student 2419):