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:
Complete free-body diagrams with forces and perpendicular lever arms, and discusses/demonstrates:
- the magnitude of the weight force w is the same for both higher and lower beams; and
- for each beam, the lever arm for the applied force F is twice the lever for the weight force w (2⋅ℓw = ℓF); and
- Newton's first law for rotations applies to both higher and lower beams, where the ccw force torque F⋅(ℓF) and cw weight torque w⋅(ℓw) must balance each other out, and so: F⋅(ℓF) = w⋅(ℓw), F = w⋅(ℓw/ℓF) = w⋅(ℓw/(2⋅ℓw)) = w/2; such that
- 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.
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 ℓF lever arm is always twice the ℓw lever arm for both situations. Instead, argues that since the ℓF and ℓw values for the higher beam are both bigger than the respective ℓF and ℓw values for the lower beam, then the higher beam F = w⋅(ℓw/ℓF) = w⋅(bigger/bigger) must be equal to the lower beam F = w⋅(ℓw/ℓF) = 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.
Nearly correct, but argument has conceptual errors, or is incomplete. As (r), but does not clearly/correctly show ℓF and ℓw 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.
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.
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.
Irrelevant discussion/effectively blank.
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):