Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Problem 25

Chapter 17, Problem 25

The pendulum in a grandfather clock is made of brass and keeps perfect time at 17°C. How much time is gained or lost in a year if the clock is kept at 26°C? (Assume the frequency dependence on length for a simple pendulum applies; see Chapter 14.)

Verified step by step guidance

Identify the relationship between the period of a pendulum and its length. The period of a simple pendulum is given by the formula:

T = 2 π \sqrt{\frac{l}{g}}

, where

T

is the period,

l

is the length of the pendulum, and

g

is the acceleration due to gravity.

Understand how the length of the pendulum changes with temperature. The length of the pendulum made of brass will expand linearly with temperature according to the formula:

l = l__{0} (1 + α ∆ T)

, where

l

is the new length,

l_{0}

is the original length,

α

is the coefficient of linear expansion for brass, and

∆ T

is the temperature change.

Determine the fractional change in the period due to the change in length. Since the period depends on the square root of the length, the fractional change in the period is approximately half the fractional change in the length:

\frac{∆}{T} \approx \frac{1}{2} \times α ∆ T

Calculate the total time gained or lost in a year. Multiply the fractional change in the period by the total number of seconds in a year:

∆ t = (\frac{1}{2} \times α ∆ T) \times 31,536,000

, where 31,536,000 is the number of seconds in a year.

Substitute the known values: the coefficient of linear expansion for brass (

α \approx 19 \times 10 ⁻ 6

), the temperature change (

∆ T = 26 - 17

), and the total seconds in a year to compute the time gained or lost. Ensure the units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Thermal Expansion

Thermal expansion refers to the increase in size or volume of materials as they are heated. In the context of the pendulum, the brass will expand when the temperature rises from 17°C to 26°C, affecting its length and, consequently, its oscillation frequency. This change in length alters the time period of the pendulum's swing, which is crucial for understanding how timekeeping is affected by temperature.

Pendulum Frequency and Length Relationship

The frequency of a simple pendulum is inversely related to the square root of its length. As the length of the pendulum increases due to thermal expansion, the frequency decreases, leading to a longer period of oscillation. This relationship is fundamental in determining how much time is gained or lost by the clock when the temperature changes.

Time Period of a Pendulum

The time period of a pendulum is the time it takes to complete one full swing back and forth. It is given by the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Understanding how the time period changes with temperature-induced length variations is essential for calculating the time gained or lost by the clock over a year.

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