67. Temperature and the Period of a Pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equation T = 27L where g is the constant acceleration of gravity at the pendulum's location. If we measure g in centimeters per second squared, we measure L in centimeters and T in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with u being temperature and k the proportion- ality constant, dL = kL. du Assuming this to be the case, show that the rate at which the pe- riod changes with respect to temperature is kT/2. See page 156.
67. Temperature and the Period of a Pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equation T = 27L where g is the constant acceleration of gravity at the pendulum's location. If we measure g in centimeters per second squared, we measure L in centimeters and T in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with u being temperature and k the proportion- ality constant, dL = kL. du Assuming this to be the case, show that the rate at which the pe- riod changes with respect to temperature is kT/2. See page 156.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Transcribed Image Text:67. Temperature and the Period of a Pendulum For
oscillations of small amplitude (short swings), we may safely
model the relationship between the period T and the length L of
a simple pendulum with the equation
T = 27L
where g is the constant acceleration of gravity at the pendulum's
location. If we measure g in centimeters per second squared, we
measure L in centimeters and T in seconds. If the pendulum is
made of metal, its length will vary with temperature, either
increasing or decreasing at a rate that is roughly proportional to
Transcribed Image Text:L. In symbols, with u being temperature and k the proportion-
ality constant,
dL
= kL.
du
Assuming this to be the case, show that the rate at which the pe-
riod changes with respect to temperature is kT/2. See page 156.
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