Calculating Surface Expansion Of A Zinc Plate Due To Temperature Variation

Hey guys! Let's dive into a fascinating physics problem today. We're going to figure out how much a square zinc plate expands when its temperature changes. This is a classic example of thermal expansion, and it's super important in lots of real-world applications, from designing bridges to building electronic devices.

Understanding Thermal Expansion

Before we jump into the calculations, let's quickly recap what thermal expansion is all about. Basically, when you heat up a material, its atoms start jiggling around more vigorously. This increased atomic motion causes the average distance between atoms to increase, which means the material expands. The amount of expansion depends on a few things: the material itself, the original size of the material, and the temperature change. Different materials expand at different rates, which is why we have something called the coefficient of thermal expansion. This coefficient tells us how much a material's size changes for every degree Celsius (or Fahrenheit) change in temperature.

Types of Thermal Expansion

There are three main types of thermal expansion:

• Linear Expansion: This is the change in length of a material. Think of a metal rod getting longer when heated.
• Area Expansion (or Surface Expansion): This is the change in the surface area of a material. Our zinc plate example falls into this category.
• Volume Expansion: This is the change in volume of a material. Imagine a balloon expanding when you heat the air inside it.

In our case, we're dealing with surface expansion, which means we need to focus on how the area of the zinc plate changes with temperature.

The Problem: Zinc Plate Expansion

Okay, let's get down to the specifics of the problem. We have a square zinc plate that's two meters on each side. This gives us an initial area to work with, which we'll need for our calculations. The temperature of the plate changes from 53 degrees Fahrenheit to 22 degrees Celsius. That's quite a swing in temperature! Our goal is to figure out how much the surface area of the zinc plate expands due to this temperature change. To solve this, we'll need to use the formula for surface expansion, which incorporates the coefficient of thermal expansion for zinc.

Key Information

Let's break down the key information we have:

• Material: Zinc
• Initial Shape: Square
• Initial Side Length: 2 meters
• Initial Temperature: 53°F
• Final Temperature: 22°C

What We Need to Find

We need to determine the change in surface area of the zinc plate. This means we'll calculate the initial area, then calculate the final area after the temperature change, and finally subtract the initial area from the final area to find the difference.

The Formula for Surface Expansion

Here's the magic formula we'll use to calculate the surface expansion:

ΔA = A₀ * β * ΔT

Where:

• ΔA is the change in area (what we're trying to find)
• A₀ is the initial area
• β is the coefficient of surface expansion
• ΔT is the change in temperature

Let's break down each part of this formula so we know exactly what we're dealing with.

Understanding the Components

• ΔA (Change in Area): This is the amount the surface area of the zinc plate increases (or potentially decreases, if the temperature goes down). It's measured in square meters (m²) in our case.
• A₀ (Initial Area): This is the original surface area of the zinc plate before the temperature change. Since it's a square, we can calculate this by simply squaring the side length (A₀ = side * side). In our problem, the side length is 2 meters, so the initial area is 2 m * 2 m = 4 m².
• β (Coefficient of Surface Expansion): This is a material property that tells us how much a material's area changes for every degree Celsius (or Kelvin) change in temperature. It's specific to the material. For zinc, the coefficient of linear expansion (α) is approximately 30 x 10⁻⁶ /°C. Since β (surface expansion) is approximately 2α, β for zinc is about 60 x 10⁻⁶ /°C. This is a crucial value we'll need to look up (or be given) for the material we're working with.
• ΔT (Change in Temperature): This is the difference between the final temperature and the initial temperature (ΔT = T_final - T_initial). However, here's a tricky bit: our temperatures are given in different units (Fahrenheit and Celsius). We need to convert them to the same unit before we can calculate the difference. Let's convert Fahrenheit to Celsius first. We will use the formula: Celsius = (Fahrenheit - 32) * 5/9.

Step-by-Step Calculation

Now, let's put everything together and calculate the surface expansion of the zinc plate. We'll go through each step carefully to make sure we get the right answer.

Step 1: Calculate the Initial Area (A₀)

As we already discussed, the initial area of the square zinc plate is simply the side length squared:

A₀ = side * side = 2 m * 2 m = 4 m²

So, our zinc plate starts with a surface area of 4 square meters.

Step 2: Convert Temperatures to the Same Unit

We have our initial temperature in Fahrenheit (53°F) and our final temperature in Celsius (22°C). To calculate the temperature change (ΔT), we need to have both temperatures in the same unit. Let's convert 53°F to Celsius using the formula:

Celsius = (Fahrenheit - 32) * 5/9

Celsius = (53 - 32) * 5/9

Celsius = 21 * 5/9

Celsius ≈ 11.67°C

Now we have both temperatures in Celsius: Initial Temperature (T_initial) ≈ 11.67°C and Final Temperature (T_final) = 22°C.

Step 3: Calculate the Change in Temperature (ΔT)

Now that both temperatures are in Celsius, we can easily calculate the change in temperature:

ΔT = T_final - T_initial

ΔT = 22°C - 11.67°C

ΔT ≈ 10.33°C

The temperature of the zinc plate increased by approximately 10.33 degrees Celsius.

Step 4: Apply the Surface Expansion Formula

Now we have all the pieces we need to use the surface expansion formula:

ΔA = A₀ * β * ΔT

We know:

• A₀ = 4 m²
• β = 60 x 10⁻⁶ /°C (Coefficient of surface expansion for zinc)
• ΔT ≈ 10.33°C

Let's plug these values into the formula:

ΔA = 4 m² * (60 x 10⁻⁶ /°C) * 10.33°C

ΔA ≈ 4 m² * 0.00006 /°C * 10.33°C

ΔA ≈ 0.0024792 m²

Step 5: Interpret the Result

The change in surface area (ΔA) is approximately 0.0024792 square meters. This means the zinc plate expanded by a very small amount due to the temperature change. To put it in perspective, this is about 2.48 square centimeters. It might not seem like much, but in many engineering applications, even small expansions can be significant, especially when dealing with large structures or precise instruments.

Conclusion: The Zinc Plate's Expansion

So, guys, we've successfully calculated the surface expansion of our zinc plate! By understanding the principles of thermal expansion and using the appropriate formula, we were able to determine that the plate expanded by approximately 0.0024792 square meters when its temperature increased from 53°F to 22°C. This exercise highlights the importance of considering thermal expansion in various real-world scenarios. Understanding these concepts is crucial for anyone interested in physics, engineering, or material science. Keep exploring and learning!

I hope this breakdown was helpful and clear. Remember, physics is all about understanding the world around us, one calculation at a time. Keep those questions coming!