Question

A set of bookshelves rests on a hard floorsurface 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 is 522,262.20 Pa (pascals).

Step-by-step solution

Calculate the force exerted by the shelves and books

To find the force exerted by the shelves and books, we can use the formula F = m * g, where m is the mass and g is the gravitational acceleration (approximately 9.81 m/s²). F = 262 kg * 9.81 m/s² = 2570.22 N So, the force exerted by the shelves and books is 2570.22 N.

Calculate the area in contact with the floor

To find the area in contact with the floor, we need to consider the dimensions of each leg and the fact that there are four legs. The cross-sectional dimensions for each leg are 3.0 cm * 4.1 cm. We need to convert this to meters to be consistent with the units for the force. Dimensions in meters: 0.03 m * 0.041 m Now, we can calculate the total area in contact with the floor by multiplying the area of one leg by 4 (since there are 4 legs). A = 4 * (0.03 m * 0.041 m) = 4 * 0.00123 m² = 0.00492 m² So, the total area in contact with the floor is 0.00492 m².

Calculate the pressure exerted by the shelf footings

Finally, we can use the formula for pressure to find the pressure exerted by the shelf footings on the surface. P = F / A = 2570.22 N / 0.00492 m² = 522262.20 Pa Therefore, the pressure exerted by the shelf footings on the surface is 522,262.20 Pa (pascals).

Key concepts

Force

Force is a fundamental concept in physics and it represents the interaction that changes the motion of an object. In every pressure calculation, determining the force is crucial because pressure is essentially the distribution of force over a specified area. Force is calculated using Newton's second law of motion, which states that the force exerted by an object is the product of its mass and the acceleration due to gravity.

In the context of the book shelves resting on the floor, the force they exert is determined by their total mass and the gravitational pull they experience. The formula to calculate force (\(F\) ) is:

• \(F = m \cdot g\)
• where \(m\) is the mass (in kilograms) and \(g\) is the gravitational acceleration (approximately \(9.81 \ m/s^2\) on Earth).

For the shelves, with a total mass of \(262 \, \mathrm{kg}\), this force is calculated to be \(2570.22 \, \mathrm{N}\) (Newtons). Understanding this step is key because it sets the stage for calculating pressure, which is the next step in the process.

Area

Area is another essential component when calculating pressure, as pressure is defined as the force applied per unit area. For accurate pressure calculations, it's vital to measure the area of contact between the object and the surface it rests upon. In the problem involving the bookshelves, the area in contact with the floor is defined by the legs of the shelves. Each leg has a cross-sectional area determined by multiplying its length by its width.

Given dimensions in centimeters, let's convert them into meters for uniformity in calculations, as the standard unit of measurement in physics is the meter.

• The original dimensions are \(3.0 \, \mathrm{cm} \times 4.1 \, \mathrm{cm}\).
• Converting to meters: \(0.03 \, \mathrm{m} \times 0.041 \, \mathrm{m}\).

The area contributed by one leg, therefore, is \(0.00123 \, \mathrm{m^2}\). Since there are four legs, the total area in contact with the floor is \(0.00492 \, \mathrm{m^2}\). Accurately calculating this area is crucial, as it directly affects the calculation of pressure.

Unit Conversion

Unit conversion is important because physics problems frequently involve measurements in different units, and using consistent units is key to getting accurate results. In this problem, converting units from centimeters to meters was essential to match the general metric units used throughout the calculations.

Converting measures involve understanding the relationship between units:

• \(1 \, \mathrm{cm} = 0.01 \, \mathrm{m}\)

Therefore, each leg's dimension (\(3.0 \, \mathrm{cm} = 0.03 \, \mathrm{m}\) and \(4.1 \, \mathrm{cm} = 0.041 \, \mathrm{m}\)) was converted to meters before proceeding with the pressure calculation.

Such conversions ensure precision and accuracy, especially essential in scientific calculations where small errors can result in significantly incorrect outcomes. Having consistent units allows seamless computation; for instance, force in Newtons (\(\mathrm{N}\)) and area in square meters (\(\mathrm{m^2}\)) to give pressure in Pascals (\(\mathrm{Pa}\)). Mastery of unit conversion is thus fundamental for effectively navigating physics problems.