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

Short answer

The pressure exerted by the shelf footings on the floor surface is approximately 523,031.50 Pa (Pascals).

Step-by-step solution

Calculate the total force exerted by the shelves and books

First, we need to find the total force exerted by the bookshelves and the books. To do this, we'll use the formula for force, which is: Force = Mass × Gravity. Given the mass of the shelves and the books: \(m = 262 kg\) The acceleration due to gravity: \(g = 9.81 m/s^2\) Force = Mass × Gravity Force = \(m \times g\) Now, plug in the values and solve for the force: Force = \(262 kg \times 9.81 m/s^2\) Force = \(2571.82 N\) So, the total force exerted by the shelves and books is approximately 2571.82 N.

Calculate the combined area of the four legs in contact with the floor

The given cross-sectional dimension of each leg is \(3.0 cm \times 4.1 cm\). To find the combined area of the four legs, we'll first calculate the area of a single leg and then multiply it by the number of legs (4). Area of 1 leg = Length × Width Area of 1 leg = \(3.0 cm \times 4.1 cm\) Since we need the area in square meters, convert the dimensions from centimeters to meters: Area of 1 leg = \((0.03 m) \times (0.041 m)\) Area of 1 leg = \(0.00123 m^2\) Now, find the combined area of the four legs: Total area = Area of 1 leg × Number of legs Total area = \(0.00123 m^2 \times 4\) Total area = \(0.00492 m^2\) The combined area of the four legs is 0.00492 \(m^2\).

Calculate the pressure exerted by the shelf footings on the floor surface

Now that we have the total force exerted by the shelves and books (2571.82 N) and the combined area of the four legs in contact with the floor (0.00492 \(m^2\)), we can use the formula for pressure to calculate the pressure exerted by the shelf footings on the floor surface. Pressure = Force / Area Plug in the values and solve for the pressure: Pressure = \(2571.82 N / 0.00492 m^2\) Pressure = \(523,031.50 Pa\) The pressure exerted by the shelf footings on the floor surface is approximately 523,031.50 Pa (Pascals).

Key concepts

Force Calculation

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 Conversion

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

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.