Physics quiz question: San Luis Obispo, CA pendulum period

Physics 205A Quiz 6, fall semester 2018
Cuesta College, San Luis Obispo, CA

the longest pendulum currently in existence is at the Oregon Convention Center in Portland, OR, a 40 kg mass swinging on a cable with a period of 9.25 seconds.[*] At that location the magnitude of the acceleration due to gravity g is 9.82611 m/s².[**]

In San Luis Obispo, CA, the magnitude of the acceleration due to gravity g is 9.80844 m/s².[***]

If a similar pendulum were constructed in San Luis Obispo, CA, and set into the motion with the same amplitude, it would have a period __________ the period of the pendulum in Portland, OR.
(A) less than.
(B) equal to.
(C) greater than.
(D) (Not enough information is given.)

[*] Martin Beech, The Pendulum Paradigm: Variations on a Theme and the Measure of Heaven and Earth, BrownWalker Press (2014), p. 42.
[**] wolframalpha.com/input/?i=gravitational+acceleration+portland,+or.
[***] wolframalpha.com/input/?i=gravitational+acceleration+san+luis+obispo,+ca.

Correct answer (highlight to unhide): (C)

The period of a pendulum is given by:

T = 2·π·√(L/g),

which does not depend on the mass, and only on the length L of the cable (which is the same for both locations), and the gravitational acceleration constant g (which is different for these locations).

Since San Luis Obispo, CA has a slightly smaller gravitational acceleration constant (9.80844 m/s²) compared to Portland, OR (9.82611 m/s²), then the pendulum in San Luis Obispo, CA will have a slightly longer period compared to Portland, OR.

This can be shown quantitatively by writing out the pendulum period equations for both locations:

T_Portland = 2·π·√(L_Portland/g_Portland),

T_SLO = 2·π·√(L_SLO/g_SLO).

Since the cable length would be the same for both locations, then:

L_SLO = L_Portland,

and solving for the period for the pendulum in San Luis Obispo, CA:

g_SLO·(T_SLO/(2⋅π)² = g_Portland·(T_Portland/(2⋅π)²,

g_SLO·(T_SLO/(2⋅π)² = g_Portland·(T_Portland/(2⋅π)²,

(T_SLO)² = (T_Portland)²⋅(g_Portland/g_SLO) ,

T_SLO = T_Portland√(g_Portland/g_SLO),

T_SLO = (9.25 s)·√((9.82611 m/s²)/(9.80844 m/s²)) = 9.2583282333... s,

which to three significant figures is 9.26 s, and thus longer than the period of the pendulum in Portland, OR.

Sections 70854, 70855
Exam code: quiz06POr7
(A) : 10 students
(B) : 10 students
(C) : 32 students
(D) : 0 students

Success level: 61%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.58