Physics quiz question: rotational kinetic energy fraction

Physics 205A Quiz 5, fall semester 2011
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 8.9

A soccer ball (which can be approximated as a thin hollow spherical shell) rolls without slipping down a ramp, after starting from rest. Neglect kinetic friction and drag. As the soccer ball rolls down the ramp, what fraction of the total kinetic energy is rotational?
(A) 0.20.
(B) 0.33.
(C) 0.40.
(D) 0.67.

Correct answer: (C)

The total kinetic energy of the soccer ball is the sum of its translational and rotational components:

K = K_tr + K_rot,

where K_tr = (1/2)*m*v², and K_rot = (1/2)*I*ω². Since the soccer ball can be approximated as a spherical shell, I = (2/3)*m*r², and for rolling without slipping, ω = v/r. Substituting these into the K_rot term:

K = (1/2)*m*v² + (1/2)*(2/3)*m*r²*(v/r)²,

K = (1/2)*m*v² + (1/3)m*v² = (5/6)*m*v².

The fraction of the total kinetic energy that is rotational kinetic energy is then:

K_rot/K = ((1/3)m*v²)/((5/6)*m*v²) = (1/3)*(6/5) = (2/5) = 0.40.

Response (A) is 2/5; response (B) is 1/3; response (D) is 2/3.

Sections 70854, 70855
Exam code: quiz05t0rQ
(A) : 0 students
(B) : 29 students
(C) : 7 students
(D) : 15 students

Success level: 14%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.23