Physics midterm question: comparing horizontal forces supporting tilted beams

Physics 205A Midterm 2, fall semester 2018
Cuesta College, San Luis Obispo, CA

A horizontal force is applied to hold a uniform beam stationary at an angle of 80° above the horizontal, and another horizontal force is applied 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 beam.) Discuss why less force required to hold the beam when it is at the higher 80° angle. 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; but
  2. the lever arms ℓ_F are not the same, where ℓ_F is longer for the higher beam, and shorter for the lower beam; and
  3. the lever arms ℓ_w are not the same, where ℓ_w is shorter for the higher beam, and longer for the lower beam; and
  4. Newton's first law for rotations applies to both higher and lower beams, where the ccw force τ = F⋅ℓ_F and cw weight τ = w⋅ℓ_w must balance each other out, and so:

     F⋅ℓ_F = w⋅ℓ_w,
     
     F = w⋅(ℓ_w/ℓ_F);

     such that
  5. for the higher beam, the shorter ℓ_w in the numerator and longer ℓ_F in the denominator means that the applied force F is smaller than for the lower beam (where it has a longer ℓ_w in the numerator and a shorter ℓ_F in the denominator).
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. At least demonstrates that for the higher beam ℓ_w is shorter and ℓ_F is longer, but typically discusses only how one of these contributes to making the applied force smaller.
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. Typically argues that F is smaller for the higher beam because ℓ_F is larger (while claiming ℓ_w is the same for both beams); or F is smaller for the higher beam because ℓ_w is smaller (while claiming ℓ_F is the same for both beams).
• 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: midterm02r3iN
p: 17 students
r: 4 students
t: 21 students
v: 10 students
x: 5 students
y: 0 students
z: 0 students

A sample "p" response (from student 5250):