Determine the number of atoms in 1 cm³ of copper.

Calculate Mass of Hollow Spherical Shell Given Inner and Outer Radius

Post Sections

• Reference
• Question
• Explain Question
• Explain Solve
• Summary
• Solve

Reference

Serway & Jewett's "Physics for Scientists and Engineers with Modern Physics 9th Edition" is a comprehensive physics textbook that covers mechanics, thermodynamics, electricity, magnetism, optics, and modern physics. The book is known for its clear explanations, detailed examples, and rigorous approach to problem-solving. This question comes from the mechanics section, specifically dealing with density and volume calculations.

Question

What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r₁ and outer radius r₂?

Explain Question

This problem involves calculating the mass of a hollow spherical shell. We are given:

• Material density (ρ)
• Inner radius (r₁)
• Outer radius (r₂)

To find the mass, we need to:

• Calculate the total volume of the shell (outer sphere minus inner sphere)
• Use the density formula to find mass

Explain Solve

Let's solve this step by step:

1. The volume of a solid sphere is V = (4/3)πr³
2. For a hollow sphere, we subtract the inner volume from the outer volume:
   V = V₂ - V₁ = (4/3)π(r₂³ - r₁³)
3. We know that density is mass divided by volume: ρ = m/V
4. Rearranging for mass: m = ρV
5. Substituting our volume equation:
   m = ρ(4/3)π(r₂³ - r₁³)
6. Simplifying:
   m = (4πρ/3)(r₂³ - r₁³)

Summary

To find the mass of a hollow spherical shell, we first calculated the volume by subtracting the inner sphere's volume from the outer sphere's volume. Then, we multiplied this volume by the material's density to find the mass. The key concept is using the difference of cubes in the volume calculation.

Solve

The mass of the hollow spherical shell is given by:
m = ρ(4/3)π(r₂³ - r₁³)
where:
- ρ is the density of the material
- r₂ is the outer radius
- r₁ is the inner radius

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