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