Physics midterm question: tension in cable lowering a boom crane

Physics 205A Midterm 2, fall semester 2017
Cuesta College, San Luis Obispo, CA

A cable anchored to a wall suspends a uniform beam, which can either be held 40° above the horizontal, or 40° below the horizontal. (Calculate all torques with respect to the pivot, located at the base of the beam.) Discuss why there is less tension in the cable when the beam is held 40° above the horizontal, compared to when it is held 40° below the horizontal. Explain your reasoning using diagram(s) with locations of forces and perpendicular lever arms, the properties of torques, and Newton's laws.

Solution and grading rubric:

• p:
  Complete free-body diagrams with forces and perpendicular lever arms, and discusses/demonstrates:
  1. for either case (40° above or below the horizontal), the ccw cable tension torque = ℓ_T⋅T and cw weight torque = ℓ_w⋅w; and
  2. since Newton's first law for rotations applies to both cases, the ccw cable tension torque equals the cw weight torque such that ℓ_T⋅T = ℓ_w⋅w; and
  3. the perpendicular lever arm ℓ_w and the weight force is the same for both cases, where ℓ_w = (L/2)·cos40° and w = m⋅g; and
  4. since the cw weight torque is the same for both cases, the ccw cable tension torque is also the same for both cases; and
  5. the first case (40° above the horizontal) has a larger perpendicular lever arm ℓ_T, such that its tension T has a smaller magnitude compared to the second case (40° below the horizontal), with a smaller perpendicular lever arm ℓ_T, such that its tension T has a greater magnitude. Must at least clearly draw these ℓ_T lever arms to compare their relative lengths.
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. May have incorrect or missing cable lever arm.
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. Has at least three of the (1)-(5) steps above.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some garbled attempt at applying Newton's first law to torques, forces, and perpendicular lever arms.
• x:
  Implementation of ideas, but credit given for effort rather than merit. Approach other than that of applying Newton's first law to torques, forces, and perpendicular lever arms.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Sections 70854, 70855
Exam code: midterm02bu2Z
p: 14 students
r: 3 students
t: 20 students
v: 10 students
x: 3 students
y: 0 students
z: 0 students

A sample "p" response (from student 2635):