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Chapter 16: Problem 77

A pendulum consists of a large weight suspended by a steel wire that is \(0.9500 \mathrm{m}\) long. (a) If the temperature increases, does the period of the pendulum increase, decrease, or stay the same? Explain. (b) Calculate the change in length of the pendulum if the temperature increase is \(150.0 \mathrm{C}^{\circ}\) (c) Calculate the period of the pendulum before and after the temperature increase. (Assume that the coefficient of linear expansion for the wire is \(12.00 \times 10^{-6} \mathrm{K}^{-1}\) and that \(g=9.810 \mathrm{m} / \mathrm{s}^{2}\) at the location of the pendulum.)

Short Answer

Expert verified
(a) The period increases with temperature. (b) Change in length: 0.00171 m. (c) Periods: Before = 1.951 s, After = 1.954 s.

Step by step solution

01

Understand the Effect of Temperature on the Period

When the temperature increases, the steel wire of the pendulum undergoes thermal expansion, which means its length increases. The period of a simple pendulum is given by the formula \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity. As \( L \) increases, the period \( T \) will also increase because \( T \) is directly proportional to the square root of \( L \).
02

Calculate the Change in Length Due to Temperature Increase

The change in length \( \Delta L \) due to a temperature change can be calculated using the formula \( \Delta L = L_{0} \alpha \Delta T \), where \( L_{0} = 0.9500 \ \mathrm{m} \) is the initial length of the wire, \( \alpha = 12.00 \times 10^{-6} \ \mathrm{K}^{-1} \) is the coefficient of linear expansion, and \( \Delta T = 150.0 \ \mathrm{C}^{\circ} \) is the change in temperature. Substituting these values, we have:\[ \Delta L = 0.9500 \times 12.00 \times 10^{-6} \times 150.0 = 0.00171 \ \mathrm{m} \]
03

Calculate the Period Before the Temperature Increase

Before the temperature increase, the length of the pendulum is \( L_{0} = 0.9500 \ \mathrm{m} \). Using the period formula:\[ T_{0} = 2\pi \sqrt{\frac{0.9500}{9.810}} \]Calculating this gives:\[ T_{0} = 2\pi \times 0.3106 = 1.951 \ \mathrm{s} \]
04

Calculate the Period After the Temperature Increase

After the temperature increase, the new length \( L \) is \( L = 0.9500 + 0.00171 = 0.95171 \ \mathrm{m} \). The new period is given by:\[ T = 2\pi \sqrt{\frac{0.95171}{9.810}} \]Calculating this gives:\[ T = 2\pi \times 0.3111 = 1.954 \ \mathrm{s} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermal Expansion
Thermal expansion is a critical concept when considering how materials respond to temperature changes. When a material like the steel wire in a pendulum is subjected to an increase in temperature, it expands. This is because heat causes the molecules within the material to move more energetically, which increases the distance between them. Therefore, when we heat the steel wire, its length will increase.
This increase in length will subsequently affect the pendulum's period. In the context of a pendulum, the period is determined by the length of the wire. Thus, the longer the pendulum, the longer it takes for a full swing. This phenomena illustrates why thermal expansion is relevant in various engineering applications, where temperature changes can affect the performance and reliability of structures.
Coefficient of Linear Expansion
The coefficient of linear expansion, denoted as \( \alpha \), is a numerical value representing how much a material will expand per degree change in temperature. For the steel wire in the exercise, \( \alpha = 12.00 \times 10^{-6} \ \mathrm{K}^{-1} \), which means for every degree Celsius increase in temperature, each meter of steel will expand by \( 12.00 \times 10^{-6} \) meters.
The linear expansion can be calculated using the formula: \( \Delta L = L_0 \alpha \Delta T \). Here, \( L_0 \) is the original length, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the temperature change. For a temperature increase of \( 150^\circ \mathrm{C} \), the wire length increase is \( 0.00171 \ \mathrm{m} \), demonstrating the direct application of this coefficient in real-world scenarios.
Acceleration Due to Gravity
The acceleration due to gravity, denoted by \( g \), plays a significant role in determining a pendulum's period. In the given exercise, \( g \) is equal to \( 9.810 \ \mathrm{m/s^2} \). This value is essential because it affects how quickly the pendulum swings back and forth.
The formula for the pendulum's period \( T \) is \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( L \) is the length of the pendulum. As gravity is constant for given location, changes in this value due to variation in altitude or geographical differences are usually negligible for classroom exercises. However, in precision engineering or specific scientific applications, even slight variations can be significant, influencing the calculation of the period for pendulums or similar oscillating systems.

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Most popular questions from this chapter

If heat is transferred to 150 g of water at a constant rate for \(2.5 \mathrm{min},\) its temperature increases by \(13 \mathrm{C}^{\circ} .\) When heat is transferred at the same rate for the same amount of time to a \(150-g\) object of unknown material, its temperature increases by \(61 \mathrm{C}^{\circ}\). (a) From what material is the object made? (b) What is the heating rate?

A layer of ice has formed on a small pond. The air just above the ice is at \(-5.4^{\circ} \mathrm{C}\), the water-ice interface is at \(0^{\circ} \mathrm{C}\), and the water at the bottom of the pond is at \(4.0^{\circ} \mathrm{C}\). If the total depth from the top of the ice to the bottom of the pond is \(1.4 \mathrm{m}\), how thick is the layer of ice? Note: The thermal conductivity of ice is \(1.6 \mathrm{W} /\left(\mathrm{m} \cdot \mathrm{C}^{\circ}\right)\) and that of water is \(0.60 \mathrm{W} /\left(\mathrm{m} \cdot \mathrm{C}^{\circ}\right)\).

Brain Power As you read this problem, your brain is consuming about 22 W of power. (a) How many steps with a height of \(21 \mathrm{cm}\) must you climb to expend a mechanical energy equivalent to one hour of brain operation? (b) A typical human brain, which is \(77 \%\) water, has a mass of \(1.4 \mathrm{kg}\). Assuming that the \(22 \mathrm{W}\) of brain power is converted to heat, what temperature rise would you estimate for the brain in one hour of operation? Ignore the significant heat transfer that occurs between a human head and its surroundings, as well as the \(23 \%\) of the brain that is not water.

The temperature at the surface of the Sun is about \(6000 \mathrm{K}\). Convert this temperature to the (a) Celsius and (b) Fahrenheit scales.

A grandfather clock has a simple brass pendulum of length L. One night, the temperature in the house is \(25.0^{\circ} \mathrm{C}\) and the period of the pendulum is 1.00 s. The clock keeps correct time at this temperature. If the temperature in the house quickly drops to \(17.1^{\circ} \mathrm{C}\) just after 10 P.M., and stays at that value, what is the actual time when the clock indicates that it is 10 A.M. the next morning?
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