Post-Lab Question: You are asked to construct a 1.00kg cube made of aluminum, and a 1.00kg cube made of brass. What side length should each cube have to produce the desired mass of 1.00kg?
The correct answer and explanation is:
Correct Answer:
To find the side length of each 1.00 kg cube:
Density of aluminum = 2.70 g/cm³ = 2700 kg/m³
Density of brass = 8.60 g/cm³ = 8600 kg/m³
Volume = Mass / Density
Since each cube is 1.00 kg, compute the volume for each and then find the cube root to get side length.
Aluminum:
Volume = 1.00 kg / 2700 kg/m³ = 0.00037037 m³
Side length = (0.00037037 m³)^(1/3) ≈ 0.0721 m or 7.21 cm
Brass:
Volume = 1.00 kg / 8600 kg/m³ = 0.00011628 m³
Side length = (0.00011628 m³)^(1/3) ≈ 0.0488 m or 4.88 cm
Explanation:
This problem involves applying the concept of density to determine the size of two materials of the same mass. Density is defined as mass per unit volume, and it helps determine how much space a certain mass of material will occupy. Materials with higher density will occupy less space than those with lower density, given the same mass.
To solve the question, begin by converting the known densities from grams per cubic centimeter to kilograms per cubic meter so that the units are consistent with the mass in kilograms. Then apply the formula:
Volume = Mass ÷ Density
This gives the volume that each cube should occupy. Because the object is a cube, all sides are equal, and volume is equal to the side length cubed:
Volume = Side³
Side = Cube root of Volume
When you calculate the cube root of each volume, the results give the required side lengths. Aluminum, being less dense, has a larger volume and therefore a longer side length. Brass, being denser, takes up less space, so its cube has a shorter side. This practical example shows how density affects the physical dimensions of objects with equal mass.