Physics midterm question: comparing vertical forces supporting tilted beams

Physics 205A Midterm 2, fall semester 2019
Cuesta College, San Luis Obispo, CA

A force F₁ pulls up at the end of a uniform beam to hold it stationary at an angle of 80° above the horizontal, and a force F₂ pulls up at the end of an identical uniform beam to hold it stationary at an angle of 10° above the horizontal. (Calculate all torques with respect to the pivot, located at the base of the beams.) Discuss why these forces 
 F₁ and F₂ have the same magnitude. 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. the magnitude of the weight force w is the same for both higher and lower beams; and
  2. for each beam, the lever arm for the applied force F is twice the lever for the weight force w (2⋅ℓ_w = ℓ_F); and
  3. Newton's first law for rotations applies to both higher and lower beams, where the ccw force torque F⋅(ℓ_F) and cw weight torque w⋅(ℓ_w) must balance each other out, and so: F⋅(ℓ_F) = w⋅(ℓ_w), F = w⋅(ℓ_w/ℓ_F) = w⋅(ℓ_w/(2⋅ℓ_w)) = w/2; such that
  4. the applied forces on the higher and lower beam must be equal in magnitude, as they are both equal to one-half of the weight of the beam.
• r:
  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 note that the ℓ_F lever arm is always twice the ℓ_w lever arm for both situations. Instead, argues that since the ℓ_F and ℓ_w values for the higher beam are both bigger than the respective ℓ_F and ℓ_w values for the lower beam, then the higher beam F = w⋅(ℓ_w/ℓ_F) = w⋅(bigger/bigger) must be equal to the lower beam F = w⋅(ℓ_w/ℓ_F) = w⋅(smaller/smaller), but only implicitly demonstrates how the "bigger/bigger" ratio is exactly equal to the "smaller/smaller" ratio by use of a scaled drawing instead of using geometry/trigonometry, etc.
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. As (r), but does not clearly/correctly show ℓ_F and ℓ_w lever arms for both situations. At least has two sets of Newton's first law for rotations, one for the higher beam and one for the the lower beam, setting the ccw torques equal to the cw torques.
• 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: midterm02sQm5
p: 3 students
r: 18 students
t: 14 students
v: 14 students
x: 3 students
y: 0 students
z: 0 students

A sample "p" response (from student 5281):