Problem 29 - University Physics with Modern Physics

Question

You are given a sample of metal and asked to determine its specific heat. You weigh the sample and find that its weight is 28.4 \(\mathrm{N}\) . You carefully add \(1.25 \times 10^{4} \mathrm{J}\) of heat energy to the sample and find that its temperature rises 18.0 \(\mathrm{C}^{\circ} .\) What is the sample's specific heat?

Step-by-Step Solution

Answer

The specific heat is approximately 239.46 J/(kg°C).

Step 1: Understand the Units and Concepts

First, recognize that you need to calculate specific heat, which uses the formula: \( q = mc\Delta T \). Here, \( q \) is the heat added (in Joules), \( m \) is the mass (in kg), \( c \) is the specific heat (in \( J/(kg \cdot °C) \)), and \( \Delta T \) is the temperature change (in °C).

Step 2: Convert Weight to Mass

Since the weight of the sample is given as 28.4 N, convert this weight into mass using the equation \( W = mg \), where \( g \) (acceleration due to gravity) is 9.8 m/s². Rearrange to find mass: \( m = \frac{W}{g} = \frac{28.4}{9.8} \approx 2.9 \) kg.

Step 3: Apply the Specific Heat Formula

Now that you have all the necessary values, plug them into the specific heat formula: \( q = mc\Delta T \). Rearrange this to solve for specific heat \( c \): \( c = \frac{q}{m \cdot \Delta T} = \frac{1.25 \times 10^{4} \text{ J}}{2.9 \text{ kg} \cdot 18 °C} \).

Step 4: Calculate the Specific Heat

Perform the calculation: \( c = \frac{1.25 \times 10^{4}}{2.9 \times 18} \approx \frac{12500}{52.2} \approx 239.46 \text{ J/(kg} \cdot °C) \).

Key Concepts

Heat Energy

In everyday life, heat energy is the energy transferred from one object to another because of a temperature difference. It's a key concept in understanding how substances change temperature when energy is added or removed. Heat energy in physics is usually measured in Joules (J), as in our exercise where the metal sample receives 12,500 J.

When heat energy is added to a substance, it can cause the substance's particles to move faster, which is often experienced as a temperature increase. This is precisely what happens in our problem. By understanding how heat energy is transferred and measured, students can better grasp the relationship between energy input and temperature changes.

• Heat energy is crucial for changing the temperature of a substance.
• Measured in Joules, it's a quantitative way to understand energy exchanges.
• Understanding heat energy helps in solving problems where temperatures and states change.

Mass Conversion

Mass conversion is essential when working with physical quantities in physics problems. When the weight of an object is given, it's typically measured in Newtons (N). However, to calculate specific heat, we need the mass in kilograms (kg). This conversion is done by dividing the weight by the acceleration due to gravity, which is approximately 9.8 m/s² on Earth.

In the exercise, the weight of the metal sample is 28.4 N. By converting this to mass, we find:\[m = \frac{W}{g} = \frac{28.4}{9.8} \approx 2.9 \text{ kg}\]Using the correct mass is vital for accurately computing the specific heat or any other property that involves mass.

• Recognize weight as force and convert it to mass.
• Use Earth's gravitational acceleration (9.8 m/s²) for conversion.
• Ensure mass is in kg for accurate calculations in physics.

Temperature Change

Temperature change (\(\Delta T\)) is another critical component in these calculations. It represents the difference in temperature before and after heat energy is added. In our problem, the metal sample's temperature rose by 18°C after adding heat. This change is used in the specific heat formula to determine how much heat energy is required to change a substance's temperature.

Temperature change helps us understand how different materials respond to energy transfer. Some materials heat up quickly, while others require more energy to change their temperature, indicating a higher specific heat.

• Temperature change is the difference in initial and final temperatures.
• It helps determine the energy needed for temperature change.
• Key in finding specific heat capacity.

Physics Problem Solving

Physics problem solving involves understanding and applying various concepts and formulas to find a solution. In our exercise, the task is to find the specific heat of a metal sample using given values. This involves several steps, including unit conversion, understanding formulas, and performing calculations.

Here's a breakdown of the process:

• Identify and understand the formula: Specific heat, \( q = mc\Delta T \)
• Convert given units when necessary, such as weight to mass.
• Plug known values into the formula and rearrange to find the unknown.
• Perform arithmetic calculations accurately.
• Draw conclusions from the computed results.

By breaking down the process into clear steps and consistently applying known formulas and concepts, solving physics problems becomes more manageable and systematic.