Problem 101 - Thomas Calculus

Question

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 $$ T=2 \pi \sqrt{\frac{L}{g}} $$ 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 proportionality constant, $$ \frac{d L}{d u}=k L $$ Assuming this to be the case, show that the rate at which the pe- tiod changes with respect to temperature is \(k T / 2\) .

Step-by-Step Solution

Answer

The rate at which the period changes with temperature is \(\frac{k}{2}T\).

Step 1: Express the change in length with temperature

We start with the equation for the change in length of the pendulum as a function of temperature: \( \frac{dL}{du} = kL \). To find \(L\) as a function of temperature, integrate this differential equation.

Step 2: Integrate the differential equation

The differential equation \( \frac{dL}{du} = kL \) is separable. Writing it as \( \frac{1}{L}dL = k \, du \), integrate both sides to find \( L = L_0 e^{ku} \), where \( L_0 \) is the initial length of the pendulum at a reference temperature.

Step 3: Differentiate the period T with respect to u

The period of the pendulum is given by \( T = 2 \pi \sqrt{\frac{L}{g}} \) and \( L = L_0 e^{ku} \). Therefore, \( T = 2 \pi \sqrt{\frac{L_0 e^{ku}}{g}} \). Differentiate this with respect to \( u \), applying the chain rule and the derivative of \( e^{ku} \).

Step 4: Apply the chain rule and simplify

The chain rule gives us \( \frac{dT}{du} = \frac{dT}{dL} \cdot \frac{dL}{du} \). First, find \( \frac{dT}{dL} = \frac{\pi}{\sqrt{gL}} \). Then, use \( \frac{dL}{du} = kL \) to find \( \frac{dT}{du} = \frac{\pi}{\sqrt{gL}} \cdot kL = \pi k \sqrt{\frac{L}{g}} = \frac{k}{2}T \).

Step 5: Finalize the expression for rate of change of T with respect to u

From our simplification, we arrived at \( \frac{dT}{du} = \frac{k}{2}T \). This matches the desired expression for the rate at which the period changes with respect to temperature.

Key Concepts

Pendulum physics

In pendulum physics, we study the motion of a pendulum, which is a swinging weight suspended from a pivot. The physics behind it tells us how it oscillates back and forth under the influence of gravity. The equation for the period of a simple pendulum is: \[ T = 2\pi \sqrt{\frac{L}{g}} \] where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the gravity constant. For small swings, this relationship holds true and helps us understand how the period depends on the length of the pendulum.

• The longer the pendulum, the longer the period or time to complete one swing.
• The effect of mass on a pendulum’s period is negligible, simplifying calculations.
• The only requirement is that the oscillation angle is small, making it a simple harmonic motion.

These basics provide essential context for more complex calculations involving pendulums.

Differential equations

Differential equations are equations that involve an unknown function and its derivatives. They are crucial for modeling how physical quantities change and interact over time. In the problem, we encountered the differential equation: \[ \frac{dL}{du} = kL \] which represents how the length of a pendulum changes with temperature. Here:

• \( L \) is a function of temperature \( u \), showing length's dependence on temperature.
• The constant \( k \) indicates the proportional rate of change of length with temperature.

These equations permit predictions and modeling of real-world phenomena, such as how changing environmental conditions affect physical systems. Solving this involves integrating both sides of the equation.

Chain rule in calculus

The chain rule is a fundamental method for finding derivatives of composite functions in calculus. When dealing with derivatives involving functions within functions, the chain rule becomes very useful. For instance, when differentiating the pendulum's period with respect to temperature, we use the chain rule as follows: \[ \frac{dT}{du} = \frac{dT}{dL} \cdot \frac{dL}{du} \] First, calculate \( \frac{dT}{dL} \) and \( \frac{dL}{du} \), then multiply them to get \( \frac{dT}{du} \). This step is crucial in understanding the rate at which the pendulum's period varies with temperature.

• This technique allows us to manage complex dependency chains in dynamic systems.
• By breaking functions into simpler parts, we can handle intricate calculations effectively.

Understanding and applying the chain rule facilitates solving many applied calculus problems.

Temperature effects on materials

Temperature influences the physical properties of materials, including their length. Most materials expand or contract when their temperature changes. This exercise focused on how a pendulum's length, made of metal, varies with temperature. With the differential equation \( \frac{dL}{du} = kL \), we learn about:

• The linear expansion of materials when heated or cooled.
• The relation between temperature change and material expansion is proportional, represented by the constant \( k \).

Materials must be chosen and engineered to predict their behavior under varying temperatures, crucial for applications in physics and engineering. Understanding these effects allows us to make adjustments and predictions for structural integrity and performance.