Problem 16 - Chemistry The Central Science

Question

A set of bookshelves rests on a hard floor surface on four legs, each having a cross-sectional dimension of \(3.0 \times 4.1 \mathrm{cm}\) in contact with the floor. The total mass of the shelves plus the books stacked on them is 262 kg. Calculate the pressure in pascals exerted by the shelf footings on the surface.

Step-by-Step Solution

Answer

The pressure exerted by the shelf footings on the surface is approximately \(522,497 \text{ Pa}\).

Step 1: Calculate the force exerted by the shelves

To find the force exerted on the surface, we use the formula \(Force = Mass \times Gravity\). The mass of the shelves is given as 262 kg, and gravity on Earth is approximately 9.81 m/s^2. Force = 262 kg × 9.81 m/s^2 = 2570.42 N

Step 2: Determine the total area of contact

The shelves are supported by four legs, each with a cross-sectional dimension of 3.0 x 4.1 cm in contact with the floor. First, we'll need to convert these dimensions to meters, as we'll be calculating the pressure in pascals. Dimension of each leg = (3.0 cm × 4.1 cm) × \(\frac{1 m}{100 cm}\) × \(\frac{1 m}{100 cm}\) = 0.03 m × 0.041 m Total area of contact for all four legs = 4 × (0.03 m × 0.041 m) = 4 × 0.00123 m^2 = 0.00492 m^2

Step 3: Calculate the pressure

Finally, we can use the formula for pressure, Pressure = \(\frac{Force}{Area}\), to find the pressure exerted by the shelf footings on the surface. Pressure = \(\frac{2570.42 N}{0.00492 m^2}\) = 522496.75 Pa The pressure exerted by the shelf footings on the surface is approximately 522,497 Pa.

Key Concepts

Force and Pressure Relationship

Understanding the relationship between force and pressure is fundamental when studying physics, particularly in mechanics. Force pertains to the push or pull acting upon an object which may cause it to move, accelerate, or change its shape. Pressure, on the other hand, is defined as the amount of force applied over a specific area.

The equation that connects force and pressure is given by:\[ Pressure = \frac{Force}{Area} \] This indicates that pressure increases as the force applied increases, or as the area over which the force is distributed decreases. For example, if you press down on a surface with your finger, the pressure is higher than if you were to press with your entire palm, even if the force exerted is the same, because the area with the finger is much smaller.

This concept is exemplified in the textbook exercise where the weight of the bookshelf, which acts as the force due to gravity, creates pressure on the floor. The importance of this pressure calculation lies in understanding how structures support the weight and how different forces distribute over an area, which is critical in fields such as engineering and architecture.

Area and Pressure Relationship

The area over which a force is distributed significantly affects the resulting pressure. The inverse relationship between area and pressure means that for a given force, an increase in the area over which the force is applied results in a lower pressure and vice versa. This principle is why sharp knives cut better than dull ones - the sharp edge applies the same force over a much smaller area, resulting in higher pressure.

In our problem, the area of contact between each leg of the bookshelf and the floor is computed. The total area is the sum of the areas of each leg in contact with the ground. The smaller this contact area, the larger the pressure exerted on the floor per leg. This relationship is not only important for static situations, such as furniture resting on a floor, but also in dynamic situations like the pressure exerted by running shoes on the ground during a sprint.

This core principle helps students understand how the distribution of weight and force can impact an object's interaction with surfaces, which is a pivotal concept for designing and analyzing structures and their stability.

Conversion of Units

In physics and engineering problems, ensuring that all measurements are in the correct units is crucial for obtaining a meaningful answer. This typically involves converting units from one system to another, such as from the metric system to the imperial system, or within the same system, like centimeters to meters.

In this exercise, the dimensions of the bookshelf legs were initially provided in centimeters. However, to calculate pressure in pascals, a unit of the International System of Units (SI), the lengths had to be converted to meters since the pascal is defined as one newton per square meter (\(N/m^2\)). The conversion is carried out by dividing the centimeter value by 100, as there are 100 centimeters in a meter. Mathematically:\[ 1 \text{ meter} = 100 \text{ centimeters} \] Understanding and applying units conversion is essential for accuracy in scientific calculations and real-world applications, allowing proper analysis, interpretation, and communication of results.