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 is approximately \(522,356 \mathrm{~Pa}\).

Step-by-step solution

Calculate the total force exerted by the shelf system on the floor

First, we need to find the total force of the shelf system. This can be found using the formula: F = m * g Where F is the force, m is the mass (in kg) and g is the acceleration due to gravity (approximately \(9.81 \mathrm{~m/s^2}\)). We have: m = 262 kg g = \(9.81 \mathrm{~m/s^2}\) Now, calculate F: F = 262 kg * \(9.81 \mathrm{~m/s^2}\) = \(2570.02 \mathrm{~N}\) The total force exerted by the shelf system on the floor is 2570.02 N.

Calculate the total area in contact with the floor

Next, we need to find the total area that is in contact with the floor by analyzing the given dimensions of each leg. Since there are 4 legs, we add the area of each footing and then, multiply the result by 4. Area of footing = (Length) * (Width) = \(3.0 \mathrm{~cm}\) * \(4.1 \mathrm{~cm}\) Convert to meters: \((3.0 \mathrm{~cm}\) * \(4.1 \mathrm{~cm}\))/\(100^2\) Total area = 4 legs * footing area Total area = 4 * (0.03 m * 0.041 m) ≈ \(0.00492 \mathrm{~m^2}\)

Calculate the pressure exerted by the shelf on the floor

Now, we can calculate the pressure exerted by the system using the formula: Pressure = Force / Area Plug in the values we have found: Pressure ≈ 2570.02 N / 0.00492 \( \mathrm{m^2}\) Pressure ≈ \(522,356 \mathrm{~Pa}\) The pressure exerted by the shelf footings on the surface is approximately 522,356 Pa.