Problem 75 - College Physics

Question

A spherical shell is formed by taking a solid sphere of radius \(20.0 \mathrm{~cm}\) and hollowing out a spherical section from the shell's interior. Assume the hollow section and the sphere itself have the same center location (that is, they are concentric). (a) If the hollow section takes up 90.0 percent of the total volume, what is its radius? (b) What is the ratio of the outer area to the inner area of the shell?

Step-by-Step Solution

Answer

(a) The radius is approximately 18.8 cm. (b) The area ratio is about 1.135.

Step 1: Find the formula for volumes of spheres

The volume of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \) where \( r \) is the radius of the sphere. For our problem, define \( R = 20.0 \mathrm{~cm} \) as the radius of the outer sphere, and let \( r \) be the radius of the inner hollow section.

Step 2: Determine the volume of the outer sphere

The volume of the outer sphere of radius \( R = 20.0 \mathrm{~cm} \) is \( V_{outer} = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (20.0)^3 \). Calculate this to get \( V_{outer} \approx 33510.32164 \mathrm{~cm}^3 \).

Step 3: Relate the volumes to find the inner radius

If the hollow section takes 90% of the total volume, then \( V_{inner} = 0.9 \times V_{outer} \). So, \( \frac{4}{3} \pi r^3 = 0.9 \times 33510.32164 \). Solve for \( r \).

Step 4: Solve for the inner hollow radius

After substituting and solving \( \frac{4}{3} \pi r^3 = 0.9 \times 33510.32164 \), you find \( r^3 = (0.9 \times 33510.32164 \times 3) / (4\pi) \). Calculating this value, \( r \approx 18.8 \mathrm{~cm} \).

Step 5: Calculate the surface areas

The outer surface area \( A_{outer} \) of a sphere with radius \( R \) is given by \( 4\pi R^2 = 4\pi (20.0)^2 \approx 5026.54824 \mathrm{~cm}^2 \). The inner surface area \( A_{inner} \) of the hollow sphere with radius \( r \) is \( 4\pi r^2 = 4\pi (18.8)^2 \approx 4430.44829 \mathrm{~cm}^2 \).

Step 6: Calculate the ratio of outer to inner area

The ratio of the outer area to the inner area is \( \frac{A_{outer}}{A_{inner}} = \frac{5026.54824}{4430.44829} \), which simplifies to approximately \( 1.1348 \).

Key Concepts

Spherical Geometry

In the realm of spherical geometry, dealing with spheres and their properties is common. A sphere is a perfectly symmetrical 3D shape, where every point on its surface is equidistant from its center.
Understanding spherical geometry involves comprehending different aspects like radii, diameters, chords, and surface areas.

• Radius: The distance from the center of the sphere to any point on its surface.
• Diameter: Twice the radius, a straight line passing through the center connecting two points on the sphere's surface.
• Concentric Spheres: Two or more spheres sharing the same center, where the larger sphere envelops the smaller one.

These concepts are crucial when examining spherical objects, especially when they are nested within each other, as in the case of spherical shells.

Volume Calculation

Calculating the volume of spherical objects is vital across various scientific disciplines. The formula for the volume of a sphere is \[ V = \frac{4}{3} \pi r^3 \]where \( r \) is the radius.
This formula emerges from integrating over the sphere, taking into account its curvature and symmetry.

• Outer Sphere: For a sphere with a radius \( R \), calculate the outer volume as \( V_{outer} = \frac{4}{3} \pi R^3 \).
• Inner Sphere: When considering a hollow inside, we use a smaller radius \( r \) for calculating its volume \( V_{inner} = \frac{4}{3} \pi r^3 \).

In exercises involving hollow spheres or spherical shells, one might solve for unknowns like the inner radius or remaining mass using ratios of volumes.

Surface Area Ratio

A sphere's surface area is an essential attribute, determined using the formula \[ A = 4 \pi r^2 \].
For shells formed by two concentric spheres, understanding the surface area ratio is useful for various applications.

• Outer Surface Area: For the outer sphere with radius \( R \), it is \( A_{outer} = 4 \pi R^2 \).
• Inner Surface Area: For the smaller, inner hollow sphere of radius \( r \), it is \( A_{inner} = 4 \pi r^2 \).

To find how the thickness influences the surface, divide the outer surface area by the inner surface area. This ratio \[ \frac{A_{outer}}{A_{inner}} \]provides insights into how much larger one surface is compared to another, crucial in engineering and materials science.

Concentric Spheres

Concentric spheres are a fascinating concept in geometry, where multiple spheres share the same center. This structure creates shells or layers within a continuous spherical form.
The cumulative volume has applications in many areas, including astrophysics and geophysics.

• Shared Center: All spheres revolve around a single central point, differing only by radius.
• Layered Structure: Concentric spheres naturally form layers, influencing how we calculate mass and energy storage.
• Mathematical Implications: Simplifies complex calculations by allowing subtraction or integration over spherical volume.

Concentric spheres serve as a basis for understanding various natural and artificial phenomena, from the Earth’s core layers to consumer electronics.