Physics midterm problem: disk rolling uphill

Physics 5A Midterm 2, Spring Semester 2008
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 1/e, Problem 8.62

[20 points.] A solid disk of mass 1.90 kg and radius 0.0500 m rolls without slipping along a horizontal surface with a translational speed of 0.240 m/s. (The rotational inertia of a disk is given by (1/2)*M*R^2.) It comes to an incline that makes an angle of 35.0 degrees with the horizontal surface. Neglecting energy losses due to friction, to what height above the horizontal surface does the disk rise on the incline? Show your work and explain your reasoning.


Solution and grading rubric:

• p = 20/20:
  Correct. Recognizes that W_nc = 0, such that when the disk has reached its highest height, all of its K_tr and K_rot has gone into U_grav. Writes out an energy balance equation 0 = delta(K_rot) + delta(K_tr) + delta(U_grav) and solves for y_f = 0.00441 m; or solves for K_tr,i and K_rot,i separately, then sets K_tot,i = K_tr,i + K_rot,i = U_grav,f to solve for y_f. Note that since the disk rolls without slipping, w_i = v_i/R.
  
• r = 16/20:
  Nearly correct, but includes minor math errors. Applies rolling without slipping condition, and recognizes both K_rot and K_tr must be accounted for separately in energy conservation.
  
• t = 12/20:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. Attempts to apply energy conservation, but one missing energy term out of K_rot, K_tr, or U_grav.
  
• v = 8/20:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. Some attempt at finding I = (1/2)*m*R^2, energy terms, Newton's laws, or applying angular momentum conservation.
  
• x = 4/20:
  Implementation of ideas, but credit given for effort rather than merit.
  
• y = 2/20:
  Irrelevant discussion/effectively blank.
  
• z = 0/20:
  Blank.
  

Grading distribution:
p: 4 students
r: 9 students
t: 12 students
v: 9 students
x: 1 student
y: 1 student
z: 0 students

A sample of a "p" response (from student 1125) is shown below:

A sample of a "t" response (from student 1239) that calculates both K_rot and K_tr, but only applies K_tr to U_grav in energy conservation:

Another "t" response sample (from student 7137) that applies only K_rot to U_grav in energy conservation:

One more "t" response (from student 1337) that also applies only K_rot to U_grav in energy conservation, with additional editorial comments: