20131123

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:

(Fthin/Athin) = Y·(∆Lthin/L),
(Fthick/Athick) = Y·(∆Lthick/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:

Fthick = 2·Fthin.

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

Athin = π·rthin2,
Athick = π·rthick2.

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

Athick = π·rthick2 = π·(2·rthin)2 = 4·π·rthin2 = 4·Athin.

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

Y/L = Y/L,

Fthin/(Athin·∆Lthin) = Fthick/(Athick·∆Lthick),

and substituting in Fthick = 2·Fthin and Athick = 4·Athin:

Fthin/(Athin·∆Lthin) = 2·Fthin/(4·Athin·∆Lthick),

1/∆Lthin = 1/(2·∆Lthick),

thus:

Lthick = (1/2)·∆Lthin,

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

1 comment:

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