Problem 60 - Physics for Scientists and Engineers with Modern Physics

Question

The diameter of the Moon is \(3480 \mathrm{~km}\). What is the volume of the Moon? How many Moons would be needed to create a volume equal to that of Earth?

Step-by-Step Solution

Answer

About 49 Moons would be needed to match Earth's volume.

Step 1: Understand the task

We need to find the volume of the Moon given its diameter and compare it to Earth's volume to find how many Moons are needed to equal the Earth's volume.

Step 2: Determine the radius of the Moon

The radius of the Moon is half of its diameter. Given the diameter is 3480 km, the radius is calculated as follows:\[\text{Radius of the Moon} = \frac{3480}{2} = 1740 \text{ km}\]

Step 3: Calculate the volume of the Moon

Use the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius. Substitute 1740 km for \( r \):\[V_{\text{Moon}} = \frac{4}{3} \pi (1740)^3\] Calculating this gives the Moon's volume as approximately 22,022,903 cubic kilometers.

Step 4: Determine the volume of Earth

The diameter of Earth is about 12742 km, thus the radius is \( \frac{12742}{2} = 6371 \) km. Use the formula for the volume of a sphere to find Earth's volume:\[V_{\text{Earth}} = \frac{4}{3} \pi (6371)^3\] Calculating this gives Earth's volume as approximately 1,083,206,916 cubic kilometers.

Step 5: Calculate the number of Moons that fit into Earth

To find out how many Moons are needed to match Earth's volume, divide Earth's volume by the Moon's volume:\[\text{Number of Moons} = \frac{V_{\text{Earth}}}{V_{\text{Moon}}} = \frac{1,083,206,916}{22,022,903} \approx 49.18\]So, it takes about 49 Moons to equal Earth's volume.

Key Concepts

Spherical Volume Formula

Volume plays a crucial role when thinking about the size of celestial objects. For spheres, like the Moon and Earth, the formula to calculate volume is very straightforward. This formula is

• \( V = \frac{4}{3} \pi r^3 \),
• where \( V \) represents the volume, and \( r \) is the radius of the sphere.

This simple formula lets you calculate the amount of space inside a spherical object quickly. Importantly, this formula applies the same way to any sphere, irrespective of size. The formula reveals that volume grows fast as the radius increases, being proportional to the cube of the radius. This means, for example, every time you double the radius of a sphere, its volume octuples (increases eightfold)! Practicing this calculation with celestial bodies, like the Earth and Moon, helps in visualizing why the volume becomes such a large number when dealing with large diameters.

Comparison of Celestial Volumes

Now let's compare two celestial bodies, the Moon and Earth. Understanding their volumes offers insight into their relative sizes. The volume of any sphere depends heavily on its radius.

• Earth's radius is 6371 km, whereas the Moon's radius is 1740 km.
• When we calculate Earth's volume using the spherical formula, we find it to be approximately 1,083,206,916 cubic kilometers.
• The Moon's volume comes out to be around 22,022,903 cubic kilometers.

These numbers might seem incredibly vast, but they highlight important aspects of the relative scale in astronomy. Notice how Earth's radius is less than four times the Moon's radius but its volume is nearly 49 times larger. This dramatic difference demonstrates how volume increases with the cube of radius, a significant factor in understanding the impressively large numbers associated with celestial objects.

Diameter and Radius Relationship

To work with spheres effectively, understanding the connection between diameter and radius is a must. Simply put, the diameter of a circle or sphere is twice its radius.

• If you know a sphere's diameter, you can find the radius by dividing the diameter by 2.
• For example, the Moon’s diameter is given as 3480 km, enabling us to calculate its radius: \( 1740 \text{ km} \).

This straightforward relationship is vital for solving many science and math problems, especially when using formulas that require the radius, like the volume formula. Remember, in any spherical measurement,

• the diameter describes the distance from one side of the sphere to the other, passing through the center,
• while the radius measures the distance from the sphere's center to its surface.

This basic relationship simplifies many calculations, turning diameter information into useful insights for further computations.