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 surface can be calculated using the following steps: 1. Calculate the area of one leg footing: \(Area = 3.0cm \times 4.1cm\) 2. Convert the area to m²: \(Area_{m^2} = \frac{Area_{cm^2}}{10,000}\) 3. Calculate the total area in contact with the floor: \(Total Area = 4 \times Area_{m^2}\) 4. Calculate the total force exerted: \(Total Force = 262 kg \times 9.81 m/s^2\) 5. Calculate the pressure exerted: \(Pressure = \frac{Total Force}{Total Area}\) By substituting the calculated values in the steps above, we can find the pressure exerted by the shelf footings on the surface in pascals (Pa).

Step-by-step solution

Calculate the Area of One Leg Footing

To find the area of one leg footing, multiply its length (3.0 cm) by its width (4.1 cm). Area of one leg footing = length × width Area of one leg footing = 3.0 cm × 4.1 cm

Convert Area to Meters Squared

Since the pressure should be calculated in pascals, the area should be in square meters. To convert square centimeters to square meters, divide the value by 10,000. \[1 m^2 = 10,000 cm^2\] Area of one leg footing (in m²) = Area of one leg footing (in cm²) / 10,000

Calculate the Total Area in Contact with the Floor

There are four legs in total, so the total area in contact with the floor can be found by multiplying the area of one leg footing (in m²) by 4. Total area = 4 × Area of one leg footing (in m²)

Calculate the Total Force Exerted

Since we are given the total mass, we need to calculate the total force exerted using gravitational force formula which is: Force = mass × gravitational acceleration. Assuming the gravitational acceleration is 9.81 m/s². Total force = mass × gravitational acceleration Total force = 262 kg × 9.81 m/s²

Calculate the Pressure Exerted

Finally, we can calculate the pressure exerted using the pressure formula: Pressure = Force / Area Pressure = Total force / Total area Once we substitute all the calculated values, we can compute the pressure exerted by the shelf footings on the surface in pascals (Pa).

Key concepts

Pressure Formula

When we talk about calculating pressure, we refer to the force applied per unit area on the surface of an object. The pressure formula is a simple yet fundamental concept in physics and engineering, encapsulated by the equation:
\[ P = \frac{F}{A} \]
where \( P \) is the pressure, \( F \) stands for the force exerted, and \( A \) denotes the area over which the force is distributed. In the context of the bookshelves problem, the pressure exerted on the floor by each leg of the shelf can be found out by dividing the total force due to the shelves' weight by the total area of contact that the legs have with the ground.

Force Calculation

To determine the force that an object exerts on a surface due to gravity, we use the following equation:
\[ F = m \times g \]
In this equation, \( m \) represents the mass of the object and \( g \) indicates the acceleration due to gravity, which, on Earth's surface, is typically taken as \( 9.81 \, \text{m/s}^2 \). By multiplying the mass of the bookshelves and the books they hold by the acceleration due to gravity, we obtain the force in newtons (N) that the bookshelves exert on the floor.

Area Conversion

Converting between different units of area is often required in physics problems to ensure consistent units when applying formulas. In the current scenario, the shelf legs' area is initially given in square centimeters (cm²), but we need it in square meters (m²) to calculate pressure in pascals (Pa). The conversion factor between these two units is that one square meter is equal to 10,000 square centimeters:
\[ 1 \, m^2 = 10,000 \, cm^2 \]
So, to convert from cm² to m², you divide the area in cm² by 10,000. It's vital to correctly convert the area so that the pressure can be calculated without error.

Gravitational Acceleration

Gravitational acceleration is the acceleration that an object experiences due to the force of gravity when falling freely in a vacuum near the Earth's surface. This is usually approximated as \( 9.81 \, \text{m/s}^2 \). This value can vary slightly depending on one's location on the planet (altitude and latitude), but for most purposes, including textbook problems, we use the standard average. Gravitational acceleration is crucial in calculating the force exerted by an object, as it is a part of the equation \( F = m \times g \), and hence is imperative for determining the pressure on a surface due to the weight of the object.