Problem 18 - Applied Physics

Question

Granite has a mass density of \(2650 \mathrm{~kg} / \mathrm{m}^{3}\). Find its weight density in \(\mathrm{lb} / \mathrm{ft}^{3}\).

Step-by-Step Solution

Answer

The weight density of granite is approximately \(1621.47 \mathrm{~lb/ft}^{3}\).

Step 1: Understand the Relationship between Mass Density and Weight Density

To find the weight density, we need to multiply the mass density by the acceleration due to gravity. The mass density (\(\rho\)) is given in kilograms per cubic meter \(\frac{\mathrm{kg}}{\mathrm{m}^3}\). The acceleration due to gravity \(g\) is \(9.81 \, \mathrm{m/s^2}\). The weight density \(w\) is defined as \(w = \rho \times g\), and is typically given in newtons per cubic meter \(\mathrm{N/m^3}\).

Step 2: Convert Mass Density to Weight Density in SI Units

Calculate the weight density in \(\mathrm{N/m^3}\) using the formula: \[ w = 2650 \, \frac{\mathrm{kg}}{\mathrm{m}^3} \times 9.81 \, \mathrm{m/s^2} \]This calculation gives the weight density in newtons per cubic meter.

Step 3: Calculate Weight Density in SI Units

Multiply the given mass density by the gravity constant:\[ w = 2650 \times 9.81 = 25960.5 \, \mathrm{N/m^3} \]The weight density of granite in newtons per cubic meter is \(25960.5 \, \mathrm{N/m^3}\).

Step 4: Conversion of Units from SI to Imperial

To convert from \(\mathrm{N/m^3}\) to \(\mathrm{lb/ft^3}\), use the conversion factor: 1 \(\mathrm{N/m^2} = 0.020885 \mathrm{lb/ft^2}\) and for volumes, 1 \(\mathrm{m^3} = 35.3147 \mathrm{ft^3}\).Therefore, 1 \(\mathrm{N/m^3} = 0.020885 \times 35.3147 = 0.06243 \mathrm{lb/ft^3}\).

Step 5: Calculate Granite's Weight Density in Imperial Units

Using the conversion factor calculated, convert the weight density:\[ \text{Weight Density in Imperial} = 25960.5 \, \mathrm{N/m^3} \times 0.06243 \, \frac{\mathrm{lb}}{\mathrm{ft^3}} = 1621.47 \, \mathrm{lb/ft^3} \]The weight density of granite in \(\mathrm{lb/ft^3}\) is approximately \(1621.47\).

Key Concepts

Mass Density of Granite

Granite is a common and dense igneous rock, with a notable mass density of approximately 2650 kilograms per cubic meter (kg/m³). Mass density is a crucial property, as it refers to the mass of material per unit volume. It indicates how heavy granite is for its size.

• Granite's mass density: The figure of 2650 kg/m³ tells us that a cubic meter of granite weighs 2650 kilograms.
• Physical implications: Higher mass density often corresponds to a more compact and solid material, making granite a preferred choice for building and construction.

Understanding the mass density helps in comprehending how much space a given mass of granite will occupy, essential for calculations in construction and material science.

Acceleration Due to Gravity

Acceleration due to gravity, symbolized as \( g \), is a constant that represents the acceleration of an object due to Earth's gravitational pull, with a standard value of 9.81 meters per second squared (m/s²). This constant is pivotal when converting mass density into weight density.

• Role in conversions: By multiplying the mass density by gravity, you get the weight density, determining how much force is exerted by a substance due to gravity.
• In scientific terms: Weight is defined as the gravitational force exerted on an object, hence weight density (N/m³) can be derived by the product of mass density (kg/m³) and gravity (m/s²).

Using the acceleration due to gravity helps convert mass measurements into weight, which is particularly helpful when working with materials like granite to ensure that structures are safe and stable.

SI to Imperial Unit Conversion

Conversion between the International System of Units (SI) and Imperial units is a fundamental skill in science and engineering. In this exercise, the task is to convert weight density from newtons per cubic meter (N/m³) to pounds per cubic foot (lb/ft³), which involves multiple conversion factors.

• Basic conversions: 1 N/m² equals 0.020885 lb/ft², and 1 m³ equals 35.3147 ft³. These factors are used to bridge the gap between metric and imperial measurements.
• Practical relevance: Engineers and architects often need to convert units to accommodate different measurement systems prevalent in varying geographic regions.

Understanding this conversion process is essential for comparing results with international studies or constructing buildings according to local unit standards. By using a stepwise approach, the hard conversion of granite's weight density into the Imperial unit system ensures precision needed for real-world applications.