Physics final exam problem: downhill rolling cylinder

Physics 205A Final Exam, fall semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 8.59

[20 points.] A 0.45 kg cylinder is released from rest and allowed to roll without slipping down a ramp. If the cylinder is released from a height of 0.80 m above the bottom of the ramp, what is the cylinder's translational speed when it reaches the bottom? Neglect drag. Show your work and explain your reasoning.

Solution and grading rubric:

• p = 20/20:
  Correct. Sets up an energy conservation equation with changes in gravitational potential energy, translational kinetic energy, and rotational kinetic energy summing to zero, and solves for v_f (also recognizing v = R·ω for rolling without slipping, and I = (1/2)·m·R² for a solid cylinder).
  
• r = 16/20:
  Nearly correct, but includes minor math errors. At least specifically recognizes that rotational kinetic energy is relevant, but somehow unable to include it in the energy conservation equation.
  
• t = 12/20:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. Neglects rotational kinetic energy, thus solving for the translational speed of a sliding (point) mass. Or neglects translational kinetic energy, thus solving for the rotational speed of a cylinder given the gravitational potential energy of a falling equivalent point mass.
  
• v = 8/20:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. As (t), but energy balance equation (with only two terms) has math errors or minor omissions.
  
• x = 4/20:
  Implementation of ideas, but credit given for effort rather than merit. Garbled attempt at calculating an energy term, or may involve kinematics.
  
• y = 2/20:
  Irrelevant discussion/effectively blank.
  
• z = 0/20:
  Blank.
  

Grading distribution:
Section 72177
p: 0 students
r: 0 students
t: 5 students
v: 2 students
x: 4 students
y: 2 students
z: 0 students

Sections 70854, 70855
p: 4 students
r: 2 students
t: 17 students
v: 5 students
x: 16 students
y: 4 students
z: 0 students

A sample "p" response (from student 5555):

Another sample "p" response (from student 9626):

A sample "r" response (from student 7895):

A sample "t" response (from student 1445):

A sample "y" response (from student 2096), with explosive results:

Another sample "y" response (from student 7676), relocating the cylinder and ramp into a zero-g environment: