Physics quiz question: Miley Cyrus' "Wrecking Ball"

Physics 205A Quiz 6, fall semester 2013
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e Conceptual Question 10.4

Miley Cyrus, "Wrecking Ball"
Terry Richardson (director)
Vevo, September 9, 2013

In a recent music video[*], pop singer Miley Cyrus sits on a (fake) demolition wrecking ball, together approximated as a 120 kg point mass hanging from a steel cable. While stationary, the steel cable stretches by 1.2 mm when 120 kg is hanging from it. If 240 kg were suspended from a steel cable of the same length with twice the radius, then it would stretch by __________ 1.2 mm.
(A) less than.
(B) exactly.
(C) more than.
(D) (Not enough information is given.)

[*] Don't bother watching it. Also the chain in the video is simplified here as a uniform cable.

Correct answer (highlight to unhide): (A)

Hooke's law for the thin and thick cables are given by:

(F_thin/A_thin) = Y·(∆L_thin/L),
(F_thick/A_thick) = Y·(∆L_thick/L),

where the Young's modulus Y and the original, unstretched length L are the same for the thin and thick cables (being both made of steel). The thick cable has twice the load of the thin cable:

F_thick = 2·F_thin.

The radii and thus the cross-sectional areas of the thin and the thick cables are also different:

A_thin = π·r_thin²,
A_thick = π·r_thick².

The thick cable has a radius twice that of the thin cable (r_thick = 2·r_thin), which will give it a cross-sectional area of four times that of the thin cable:

A_thick = π·r_thick² = π·(2·r_thin)² = 4·π·r_thin² = 4·A_thin.

Then setting the ratio of Y/L for the thin and thick cables equal to each other:

Y/L = Y/L,

F_thin/(A_thin·∆L_thin) = F_thick/(A_thick·∆L_thick),

and substituting in F_thick = 2·F_thin and A_thick = 4·A_thin:

F_thin/(A_thin·∆L_thin) = 2·F_thin/(4·A_thin·∆L_thick),

1/∆L_thin = 1/(2·∆L_thick),

thus:

∆L_thick = (1/2)·∆L_thin,

such that the thick cable will stretch less than the thin cable.

Sections 70854, 70855, 73320
Exam code: quiz06wR3k
(A) : 12 students
(B) : 32 students
(C) : 20 students
(D) : 0 students

Success level: 19%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.08