Determine the number of atoms in 1 cm³ of copper.

Two spheres are cut from a certain uniform rock. One has radius 4.50 cm. The mass of the other is five times greater. Find its radius.

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Reference

Serway & Jewett's "Physics For Scientists And Engineers With Modern Physics 9ed" is a comprehensive textbook that covers fundamental physics principles. This renowned work includes detailed coverage of mechanics, thermodynamics, waves, and modern physics, making it an essential resource for science and engineering students.

Question

Two spheres are cut from a certain uniform rock. One has radius 4.50 cm. The mass of the other is five times greater. Find its radius.

Explain Question

Given information:

• Two spheres are cut from the same uniform rock (same density)
• First sphere radius (r₁) = 4.50 cm
• Second sphere mass (m₂) = 5 × First sphere mass (m₁)

Required: Find the radius of the second sphere (r₂)

Explain Solve

Step 1: Express the volume of a sphere using the formula V = (4/3)πr³

Step 2: Express mass using density and volume: m = ρV, where ρ is density

Step 3: Write equations for both spheres:

m₁ = ρ × (4/3)πr₁³

m₂ = ρ × (4/3)πr₂³

Step 4: Divide m₂ by m₁ to eliminate density and constants

Step 5: Solve for r₂ using the mass ratio of 5

Summary

This problem demonstrates the relationship between mass and volume for objects of the same density. By using the ratio of masses and the known radius of one sphere, we can determine the unknown radius through the relationship between volume and radius.

Solve

For either sphere:

Volume: V = (4/3)πr³

Mass: m = ρV = ρ(4/3)πr³

Dividing equations:

m₂/m₁ = r₂³/r₁³ = 5

Therefore:

r₂ = r₁ × ∛5

r₂ = 4.50 cm × ∛5

r₂ = 7.69 cm

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