Problem 26 - Physics

Question

A gas fills the right portion of a horizontal cylinder whose radius is \(5.00 \mathrm{~cm} .\) The initial pressure of the gas is \(1.01 \times 10^{5} \mathrm{~Pa}\). A frictionless movable piston separates the gas from the left portion of the cylinder, which is evacuated and contains an ideal spring, as the drawing shows. The piston is initially held in place by a pin. The spring is initially unstrained, and the length of the gas-filled portion is \(20.0 \mathrm{~cm}\). When the pin is removed and the gas is allowed to expand, the length of the gas-filled chamber doubles. The initial and final temperatures are equal. Determine the spring constant of the spring.

Step-by-Step Solution

Answer

The spring constant is 1980 N/m.

Step 1: Understand Initial and Final Conditions

Initially, the gas is confined to a volume within a cylinder of radius 5.00 cm and length 20.0 cm. The gas pressure is \(1.01 \times 10^{5}\, \mathrm{Pa}\). The gas expands to double the initial length when the pin is removed. The piston and spring start at equilibrium, with the spring unstrained initially. The temperature remains constant throughout, hinting at an isothermal process.

Step 2: Calculate Initial Volume

The initial volume of the gas, \( V_i \), can be calculated using the cylinder volume formula \( V = \pi r^2 h \) where \( r = 0.05 \) m (5 cm), and \( h = 0.2 \) m (20 cm). Thus, \( V_i = \pi (0.05)^2 (0.2) = 1.57 \times 10^{-3} \) m³.

Step 3: Determine Final Volume

The gas expands to double its initial length, so the final length is 40.0 cm or 0.4 m. The final volume is \( V_f = \pi r^2 \times 0.4 \) m. So, \( V_f = \pi (0.05)^2 (0.4) = 3.14 \times 10^{-3} \) m³.

Step 4: Apply Boyle's Law for Pressure Calculations

Since the process is isothermal, Boyle's Law \( P_i V_i = P_f V_f \) holds. Given \( P_i = 1.01 \times 10^{5} \) Pa, solving for \( P_f \) yields \[ P_f = \frac{P_i V_i}{V_f} = \frac{1.01 \times 10^{5} \times 1.57 \times 10^{-3}}{3.14 \times 10^{-3}} = 5.05 \times 10^{4} \] Pa.

Step 5: Calculate Spring Force and Extension

The change in pressure results in a change in force exerted by the piston. This force change stretches the spring. The initial force on the piston is \( F_i = P_i A \) and final force is \( F_f = P_f A \), where \( A = \pi r^2 \) is the piston area. So, \( A = \pi (0.05)^2 = 7.85 \times 10^{-3} \) m². Thus, \[ F_i = 1.01 \times 10^{5} \times 7.85 \times 10^{-3} = 793 \] N and \[ F_f = 5.05 \times 10^{4} \times 7.85 \times 10^{-3} = 397 \] N. The net force stretching the spring is \( F_s = F_i - F_f = 793 - 397 = 396 \) N.

Step 6: Use Hooke's Law to Find Spring Constant

The extension of the spring is the change in length of the gas-filled chamber, \( 0.4 - 0.2 = 0.2 \) m. According to Hooke's Law, \( F_s = k \cdot \Delta x \), where \( k \) is the spring constant and \( \Delta x = 0.2 \) m. Solving for \( k \), we find \[ k = \frac{F_s}{\Delta x} = \frac{396}{0.2} = 1980 \] N/m.

Key Concepts

Boyle's Law

Boyle's Law is a fundamental principle in the study of gases, particularly under conditions where temperature remains constant, known as isothermal conditions. It states that the pressure of a given amount of gas is inversely proportional to its volume, provided the temperature stays constant. Mathematically, this is expressed as: \[ P_i V_i = P_f V_f \] Where:

• \( P_i \) and \( V_i \) are the initial pressure and volume
• \( P_f \) and \( V_f \) are the final pressure and volume

For instance, if a gas expands and its volume doubles as in the exercise, Boyle's Law can help determine how the pressure changes. By rearranging the formula to solve for final pressure, you can gain insight into the relationship between initial and final states. This is particularly useful in solving physics problems that involve piston-cylinder setups, similar to the scenario in the original exercise.

Isothermal Process

An isothermal process is characterized by a constant temperature throughout the transformation of the system. For ideal gases, this entails that any heat input or output is balanced by work done by or on the gas, maintaining a steady temperature.
In practical terms, during an experiment like expanding gas in a cylinder, an isothermal process implies that despite volume changes, the system's thermal energy remains unchanged. This is critical because it simplifies the analysis to focus solely on pressure and volume relationships as dictated by Boyle's Law.
Understanding isothermal processes helps in visualizing how temperature constraints affect gas behavior and transformations. It is a cornerstone concept in thermodynamics, widely used in identifying processes in engines, refrigerators, and other systems involving gas expansions or compressions.

Cylinder Volume Calculation

In physics and engineering, understanding the volume of a cylinder is essential, particularly for applications involving gases and pistons. The volume of a cylinder can be calculated using the formula: \[ V = \pi r^2 h \] Where:

• \( V \) is the volume
• \( r \) is the radius
• \( h \) is the height or length of the cylinder

In scenarios where a piston moves within a cylinder, the volume changes as the height (or length) changes. For example, in our exercise, initially the gas-filled portion had a specific length. After expansion, the length doubled, and hence the volume doubled as well. Calculating accurate volumes is crucial in applying other gas laws like Boyle's Law effectively, as it underpins calculating changes in pressure or other transformations in a system.

Hooke's Law

Hooke's Law describes the behavior of springs and is one of the most straightforward concepts in mechanics. It states that the force required to extend or compress a spring is proportional to the displacement of the spring from its equilibrium position. This is expressed as: \[ F = k \cdot \Delta x \] Where:

• \( F \) is the force exerted by the spring
• \( k \) is the spring constant, a measure of the spring's stiffness
• \( \Delta x \) is the change in length of the spring from its original position

In problems that involve pistons and springs, like in our exercise, Hooke's Law allows you to calculate the spring constant once you know the force exerted on the piston and the displacement of the spring. Understanding how springs react to forces is essential in designing systems and solving mechanical problems where elastic potential energy is involved.