One cubic meter (1.00 m^3) of aluminum has a mass of 2.70 ×10^3 kg, and 1.00 m^3 of iron has a m

step 1 In this problem we are given the volume of aluminum is equal to 1 .00 meter cube and the mass of

One cubic meter (1.00 m^3) of aluminum has a mass of 2.70...

One cubic meter $\left(1.00 \mathrm{m}^{3}\right)$ of aluminum has a mass of $2.70 \times 10^{3} \mathrm{kg},$ and 1.00 $\mathrm{m}^{3}$ of iron has a mass of $7.86 \times 10^{3} \mathrm{kg}$ . Find the radius of a solid aluminum sphere that will balance a solid iron sphere of radius 2.00 $\mathrm{cm}$ on an equal-arm balance.

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Key Concepts

Density

Density is defined as the mass per unit volume of a substance. It is a fundamental property that helps determine how much mass is contained in a given volume, and it plays a crucial role in solving problems where mass, volume, and material properties intersect.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)?r³, where r is the radius. This geometric formula is essential when determining the mass of a spherical object from its density, as it accurately relates the sphere’s size to its contained volume.

Mass Equivalence in Balancing

In a balance problem, particularly one involving an equal-arm balance, the objects will balance when their masses are equal. This principle allows us to set up equations equating the masses (density times volume) of different objects, regardless of their material or shape.

Scaling Relationships

Scaling relationships in geometry refer to how changes in one dimension, such as radius, affect other properties like volume and mass. Since the volume of a sphere scales with the cube of its radius, small changes in the radius can result in significant changes in mass, an important consideration in balancing problems.

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