Physics final exam question: normal force on truck going over hill

Physics 205A Final Exam, fall semester 2013
Cuesta College, San Luis Obispo, CA

A truck drives over the top of a hill that has a circular radius, without losing contact with the hill. This truck then drives over the top of the same hill with a slightly faster speed, but it is still in contact with the hill. Discuss 
why the magnitude of the upwards normal force of the hill on the truck is smaller when the truck is going faster over the hill. Explain your reasoning by using free-body diagram(s), the properties of forces and Newton's laws.

Solution and grading rubric:

• p:
  Correct. Understands that the truck has two vertical forces acting on it:

  Weight force of Earth on truck (downwards, magnitude w = m·g).
  Normal force of road on truck (upwards, magnitude varies but must be less than w).

  For either driving fast or slow over the hill, because the truck is (momentarily) undergoing uniform circular motion as it drives over the top of the hill, then Newton's second law applies to the motion of the truck (speed is constant but direction of the velocity vector is changing), and the net force must point downwards (towards the center of the circular hill). This means that the upwards normal force must have a magnitude less than the magnitude of the weight force. Then compares driving fast over the top of the hill versus driving slow over the top of the hill:
  • the net force on the truck has a magnitude given by m·v²/r that is greater for the case of driving faster over the hill, and has a smaller magnitude for driving slower over the hill (as m and r are the same for either case);
  • the downwards weight force on the truck is constant, as it does not depend on how fast or slow the truck drives at the top of the hill;
  • the upwards normal force must be smaller for the faster speed case, in order to result in a greater downwards net force.
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes.
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some attempt at analyzing forces, free-body diagrams, and Newton's laws.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Approach does not substantively utilize forces, free-body diagrams, and Newton's laws.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.
  

Grading distribution:
Sections 70854, 70855, 73320
Exam code: finaln0M3
p: 2 students
r: 0 students
t: 16 students
v: 16 students
x: 24 students
y: 2 students
z: 1 student

A sample "p" response (from student 0618):
A sample "x" response (from student 0419):
Another sample "x" response (from student 1969):
And another sample "x" response (from student 1994):
Yet another sample "x" response (from student 2999):
And one more sample "x" response (from student 7667):