Problem 16 - Chemistry The Central Science In Si Units

Question

A set of bookshelves rests on a hard floor surface on four legs, each having a cross-sectional dimension of \(4.0 \times 5.0 \mathrm{~cm}\) in contact with the floor. The total mass of the shelves plus the books stacked on them is \(200 \mathrm{~kg}\). Calculate the pressure in atmospheres exerted by the shelf footings on the surface.

Step-by-Step Solution

Answer

The pressure exerted by the shelf footings on the surface is approximately 2.42 atm.

Step 1: Calculate the Area of One Leg

Each leg of the bookshelf has dimensions of \(4.0 \times 5.0 \text{ cm}^2\). First, convert these dimensions to meters: \(4.0 \text{ cm} = 0.04 \text{ m}\) and \(5.0 \text{ cm} = 0.05 \text{ m}\). The area \(A\) of one leg is \(A = 0.04 \text{ m} \times 0.05 \text{ m} = 0.002 \text{ m}^2\).

Step 2: Calculate Total Area in Contact with the Floor

The bookshelf has four legs, so the total area \(A_{total}\) in contact with the floor is \(A_{total} = 4 \times 0.002 \text{ m}^2 = 0.008 \text{ m}^2\).

Step 3: Find the Force Exerted by the Bookshelf (Weight)

The weight \(F\) exerted by the bookshelf can be found using the formula \(F = mg\), where \(m = 200 \text{ kg}\) is the mass and \(g = 9.81 \text{ m/s}^2\) is the acceleration due to gravity. Thus, \(F = 200 \text{ kg} \times 9.81 \text{ m/s}^2 = 1962 \text{ N}\).

Step 4: Calculate the Pressure in Pascals

Pressure \(P\) is defined as the force per unit area. Use the formula \(P = \frac{F}{A_{total}}\). Substitute the values to get \(P = \frac{1962 \text{ N}}{0.008 \text{ m}^2} = 245250 \text{ Pa} (N/m^2)\).

Step 5: Convert Pressure from Pascals to Atmospheres

1 atmosphere is equivalent to \(101325 \text{ Pa}\). The pressure in atmospheres can be calculated as \(P_{atm} = \frac{245250 \text{ Pa}}{101325 \text{ Pa/atm}} \approx 2.42 \text{ atm}\).

Key Concepts

Unit Conversion

Unit conversion is an essential skill in physics and engineering, helping us switch between different measurement systems. For this exercise, we need to convert dimensions from centimeters to meters to find the contact area of the bookshelf legs.

Remember, converting centimeters to meters involves moving from a smaller to a larger unit. You need to divide by 100, since 1 meter is 100 centimeters. So, the conversion steps are:

• 4 centimeters = 0.04 meters
• 5 centimeters = 0.05 meters

Understanding unit conversion allows for accurate calculations and helps in applying formulas correctly, as most scientific calculations use meters, the SI unit for length.

Area Calculation

To calculate the area of a rectangle, multiply its length by its width. This simple formula helps us find how much surface area is in contact with the floor.

For one bookshelf leg, use:

• Length: 0.04 meters
• Width: 0.05 meters
• Area of one leg = 0.04 m × 0.05 m = 0.002 m²

Since the shelf rests on four legs, the total area in contact will be four times the area of one leg:

• Total area = 4 × 0.002 m² = 0.008 m²

Careful area calculation is crucial, as it directly affects the resulting pressure when combined with force.

Force and Weight

Understanding force and weight is crucial in physics. Force, often due to weight, is the result of mass under the influence of gravity. It is calculated using the formula, \( F = mg \), where:

• \( m \) is mass in kilograms
• \( g \) is gravitational acceleration, typically \( 9.81 \text{ m/s}^2 \)
• \( F \) is the force in newtons (N)

For our bookshelf, with a total mass of 200 kg, the force exerted is:

• Force = 200 kg × 9.81 m/s² = 1962 N

Such a calculation is important because pressure is directly derived from this force when distributed over an area.

Pressure in Atmospheres

Pressure is defined as force distributed over an area, \( P = \frac{F}{A} \). For this problem, it measures how much force each square meter of floor area experiences.

We calculate pressure in pascals (Pa) first:

• Using the force of 1962 N and total area 0.008 m², the pressure \( P \) = \( \frac{1962 \text{ N}}{0.008 \text{ m}^2} = 245250 \text{ Pa} \) (Pascals)

Lastly, convert this pressure to atmospheres, since 1 atm = 101325 Pa:

• Pressure in atmospheres: \( P_{atm} = \frac{245250 \text{ Pa}}{101325 \text{ Pa/atm}} \approx 2.42 \text{ atm} \)

Understanding these conversions and calculations shows how scientific and practical problem-solving in physics works, especially when using real-world units like the atmosphere.