Question

A set of bookshelves rests on a hard floor surface on four legs, each having a cross-sectional dimension of \(4.0 \times 5.0 \mathrm{~cm}\) in contact with the floor. The total mass of the shelves plus the books stacked on them is \(200 \mathrm{~kg}\). Calculate the pressure in atmospheres exerted by the shelf footings on the surface.

Short answer

The pressure exerted by the shelf footings on the surface can be calculated using the following steps: the total weight of the shelves and books is 200 kg × 9.81 m/s², the total area of the four legs is 4 × \( \frac{20.0}{10000} \) m², the pressure in Pascals is (200 kg × 9.81 m/s²) / (4 × \( \frac{20.0}{10000} \) m²), and finally, the pressure in atmospheres is [(200 kg × 9.81 m/s²) / (4 × \( \frac{20.0}{10000} \) m²)] / 101325.

Step-by-step solution

Find the total weight of the shelves and books

To find the total weight, we will use the formula Weight = Mass × Acceleration due to gravity, where the mass is given in kg, and the acceleration due to gravity is approximately 9.81 m/s². Mass of shelves and books = 200 kg Acceleration due to gravity = 9.81 m/s² Weight = Mass × Acceleration due to gravity Weight = 200 kg × 9.81 m/s²

Calculate the total area of the four legs

Next, we need to calculate the total area in contact with the floor by multiplying the cross-sectional dimensions of each leg and then multiplying the result by the number of legs. Dimensions of each leg: 4.0 cm × 5.0 cm = 20.0 cm² Area of one leg = 20.0 cm² Now let's convert this area from cm² to m² to keep our units consistent (1 m² = 10,000 cm²): Area of one leg = \( \frac{20.0}{10000} \) m² There are 4 legs, so the total area of the four legs is: Total area = 4 × Area of one leg Total area = 4 × \( \frac{20.0}{10000} \) m²

Calculate the pressure exerted by the footings in Pascals

Now that we have the weight and the total area, we can calculate the pressure using the formula Pressure = Force / Area. Pressure in Pascals = Weight / Total area Pressure in Pascals = (200 kg × 9.81 m/s²) / (4 × \( \frac{20.0}{10000} \) m²)

Convert the pressure to atmospheres

To convert the pressure in Pascals to atmospheres, we will use the conversion factor 1 atmosphere = 101325 Pa. Pressure in atmospheres = Pressure in Pascals / 101325 Pressure in atmospheres = [(200 kg × 9.81 m/s²) / (4 × \( \frac{20.0}{10000} \) m²)] / 101325 Now, we can calculate the pressure in atmospheres exerted by the shelf footings on the surface.

Key concepts

Weight Calculation

To begin our pressure calculation, we first need to determine the weight exerted by the bookshelves and the books placed on them. Weight is the force acting on an object due to gravity. The formula we use is:

\[ \text{Weight} = \text{Mass} \times \text{Acceleration due to gravity} \]
The mass of the shelves and books is given as 200 kg. The standard acceleration due to gravity is approximately 9.81 m/s². So, the weight calculation for the shelves and books is:

• Mass = 200 kg
• Acceleration due to gravity = 9.81 m/s²
• Weight = 200 kg × 9.81 m/s² = 1962 Newtons (N)

Understanding weight is fundamental, as it's directly linked to the force exerted by an object due to the Earth’s gravity. This force is what we’re focusing on when calculating our pressure.

Surface Area

Surface area is crucial for distributing the weight of an object and is a key factor in calculating pressure. In this exercise, each leg of the bookshelves has a cross-sectional dimension of 4.0 cm by 5.0 cm. To find the surface area of one leg, we multiply these dimensions:

\[ \text{Area of one leg} = 4.0 \text{ cm} \times 5.0 \text{ cm} = 20.0 \text{ cm}^2 \]
However, pressure calculations usually require the area in meters squared (m²) to maintain unit consistency. Converting square centimeters to square meters requires dividing by 10,000 (since 1 m² = 10,000 cm²):
\[ \text{Area of one leg in m}^2 = \frac{20.0}{10,000} = 0.002 \text{ m}^2 \]
With four legs, we multiply the area of one leg by four:
\[ \text{Total area for four legs} = 4 \times 0.002 \text{ m}^2 = 0.008 \text{ m}^2 \]
Knowledge of how objects distribute their weight across the surface is important for practical applications like the stability of structures.

Units Conversion

To ensure accurate calculations and apply consistent measurements, we need to convert units when necessary. In this exercise, converting units came into play during the surface area calculation.
To calculate our pressure correctly, it was essential to convert:

• The surface area from cm² to m²

Conversion factors are crucial tools used in physics to seamlessly solve problems, providing coherence among different measurement units.
In this case:

• 1 m² = 10,000 cm²
• Use the conversion: \( \text{area in} \text{ m}^2 = \frac{\text{area in cm}^2}{10,000} \)

Mastering unit conversions is vital for maintaining accuracy and understanding the relationships between different measurement systems. Ensuring you're working with the correct units can dramatically impact the outcome of your calculations.

Pressure in Atmospheres

Pressure, the force per unit area, can be measured in various units, with Pascals and atmospheres being common. In this task, once we calculated pressure in Pascals (Pa), we converted it to atmospheres (atm) for ease of interpretation.

First, using the formula:
\[ \text{Pressure} = \frac{\text{Weight}}{\text{Total area}} = \frac{1962 \text{ N}}{0.008 \text{ m}^2} = 245,250 \text{ Pa} \]
Pressure in Pascals is then converted to atmospheric pressure using the conversion:
1 atmosphere = 101325 Pa
\[ \text{Pressure in atmospheres} = \frac{245,250}{101325} \approx 2.42 \text{ atm} \]
Understanding pressure conversion is valuable in various scientific and engineering disciplines, as it facilitates the expression of pressure in units more relevant to certain applications, such as atmospheric pressure measurements in meteorology.