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 \(262 \mathrm{~kg} .\) Calculate the pressure in pascals exerted by the shelf footings on the surface.

When we talk about force calculation, we're referring to the process of determining the strength of a push or pull upon an object. In physics, this is critical because force directly relates to motion and stability of structures. To calculate force, the fundamental equation we use is Newton's second law of motion:

Force (F) = Mass (m) × Acceleration (a)

In the given problem, the force is the weight of the bookshelves and the books, which is the mass times the acceleration due to gravity (9.81 m/s², the standard on Earth). To ensure accuracy, it's important to note units. Gravity's acceleration has units of meters per second squared, so mass must be in kilograms to result in the force being measured in newtons (N), which are the standard units in the International System (SI).
The calculation would look like this:
F = 262 kg × 9.81 m/s² = 2571.82 N

The result tells us that the bookshelves and the books exert a force of approximately 2571.82 N on their supports, which then apply that force onto the surface they rest upon.

Area measurement is crucial in many aspects of science and engineering, particularly when dealing with pressure. To calculate pressure, we must know the area over which the force is distributed. However, different regions may use different units to measure area, leading to area conversion being a common task.

For our bookshelf example, the leg dimensions are initially given in centimeters squared, but since the SI unit for area is square meters, we need to convert. Since 1 meter equals 100 centimeters, we convert by dividing our centimeter measurements by 100.

So for one leg, we translate the area from (3.0 cm × 4.1 cm) into square meters:
Area of 1 leg = (3.0 cm / 100) × (4.1 cm / 100) = 0.03 m × 0.041 m = 0.00123 m²

Multiplying this by the number of legs (4), we get the total area in contact with the floor, which is essential for calculating pressure.

Pascal's Law, or the Principle of Transmission of Fluid-Pressure, is a principle crucial to understanding how pressures work within confined fluids. Although our bookshelf question primarily addresses solid mechanics, Pascal's Law offers valuable insight for interpreting the problem.

The law states that pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid such that the pressure variations (initial differences) remain the same. This principle can be translated into an equation:
Pressure (P) = Force (F) / Area (A)

Applying this to the bookshelf scenario, once we've calculated the force exerted by the bookshelves and their area of contact with the ground, we can determine the pressure they exert. By dividing the force (2571.82 N) by the total area (0.00492 m²), we arrive at the pressure in Pascals (Pa), leading to our final result, which is approximately 523,031.50 Pa. Understanding Pascal's Law helps explain why, for instance, the shape of the shelf's legs or the distribution of books doesn't impact the pressure calculation—it's the total force and the total area that matter.