Problem 13 - University Physics with Modern Physics

Question

Titania, the largest moon of the planet Uranus, has \(\frac{1}{8}\) the radius of the earth and \(\frac{1}{1700}\) the mass of the earth. (a) What is the acceleration due to gravity at the surface of Titania? (b) What is the average density of Titania? (This is less the density of rock, which is one piece of evidence that Titania is made primarily of ice.)

Step-by-Step Solution

Answer

Titania's surface gravity is about 0.367 m/s² and its average density is around 1310 kg/m³.

Step 1: Understand the Problem

We need to find the acceleration due to gravity on Titania's surface and its average density. We know Titania's radius is \( \frac{1}{8} \) that of Earth and its mass is \( \frac{1}{1700} \) that of Earth.

Step 2: Formula for Surface Gravity

The formula for the acceleration due to gravity on a celestial body is given by \( g = \frac{G M}{R^2} \), where \( G \) is the gravitational constant, \( M \) is the mass of the celestial body, and \( R \) is its radius.

Step 3: Calculate Titania's Gravity

Earth's gravity \( g_{\text{Earth}} \) is approximately \( 9.8 \, \text{m/s}^2 \), mass \( M_{\text{Earth}} \), and radius \( R_{\text{Earth}} \). For Titania, \( M_{\text{Titania}} = \frac{M_{\text{Earth}}}{1700} \) and \( R_{\text{Titania}} = \frac{R_{\text{Earth}}}{8} \). Plug these into the gravity formula to get:\[g_{\text{Titania}} = \frac{G \cdot \frac{M_{\text{Earth}}}{1700}}{(\frac{R_{\text{Earth}}}{8})^2} \Rightarrow g_{\text{Titania}} = \frac{9.8 \cdot 8^2}{1700} \approx 0.367 \, \text{m/s}^2\]

Step 4: Formula for Density

The average density \( \rho \) of a celestial body is given by \( \rho = \frac{M}{\frac{4}{3}\pi R^3} \). This formula relates mass, volume, and density.

Step 5: Calculate Titania's Density

Using \( M_{\text{Titania}} = \frac{M_{\text{Earth}}}{1700} \) and \( R_{\text{Titania}} = \frac{R_{\text{Earth}}}{8} \), substitute into the density formula:\[\rho_{\text{Titania}} = \frac{\frac{M_{\text{Earth}}}{1700}}{\frac{4}{3}\pi \left(\frac{R_{\text{Earth}}}{8}\right)^3} \Rightarrow \rho_{\text{Titania}} = \frac{1700 \cdot \rho_{\text{Earth}}}{8^3} = \frac{1700}{512} \rho_{\text{Earth}} \approx 0.238 \rho_{\text{Earth}}\]Given that Earth's average density is about \( 5515 \, \text{kg/m}^3 \), Titania's density is approximately \( 1310 \, \text{kg/m}^3 \).

Step 6: Conclusion

The acceleration due to gravity on Titania is approximately \( 0.367 \, \text{m/s}^2 \), and its average density is about \( 1310 \, \text{kg/m}^3 \). Titania's low density relative to Earth's suggests it is composed largely of ice.

Key Concepts

average density calculation

To find the average density of a celestial body like Titania, we use a simple formula that relates its mass and volume. The formula for density \( \rho \) is given by:\[\rho = \frac{M}{\frac{4}{3}\pi R^3}\]where \( M \) is the mass of the object and \( R \) is its radius.
The term \( \frac{4}{3}\pi R^3 \) calculates the volume of a sphere, which is the shape we often assume for moons and planets.
This equation helps us determine how much mass is packed into the volume of the celestial body, giving us the average density.

• First, figure out the mass and radius of the celestial object in comparison to Earth if data is given that way. For Titania, its mass is \( \frac{M_{\text{Earth}}}{1700} \) and its radius is \( \frac{R_{\text{Earth}}}{8} \).
• Next, substitute these values into the formula. The radius has to be cubed because we're calculating volume.

Calculating Titania's density in this way gives insights into its composition, suggesting a lower density primarily due to a higher ice content compared to rocky counterparts like Earth.

Titania moon characteristics

Titania, as Uranus's largest moon, has some unique features. Understanding these characteristics can help us better appreciate other celestial bodies in our solar system.

• **Size and Scale**: Titania has a radius about \( \frac{1}{8} \) of Earth's radius, making it significantly smaller than our planet. This detail is crucial when studying its gravitational pull and density.
• **Mass**: With a mass that is \( \frac{1}{1700} \) of Earth's, Titania is much lighter. This means the gravitational force on its surface is also much weaker compared to Earth's. As detailed earlier, gravity here is about \( 0.367 \, \text{m/s}^2 \), which is just a fraction of Earth's \( 9.8 \, \text{m/s}^2 \).

The characteristics of Titania help to explain its scientific importance, especially in helping scientists understand smaller, icy celestial bodies in the solar system. These features contrast sharply with more massive, rocky planets and add depth to our knowledge about planet formation and composition.

celestial body density

Density is a key property when exploring celestial bodies such as moons and planets. It tells us a lot about their composition and structure.

• **What Density Reveals**: Higher density is often indicative of a rocky composition, like Earth and Venus. Low density, as observed in Titania, points towards a body made up largely of lighter materials, such as ice or gaseous components.
• **Calculation Insights**: By calculating density accurately, scientists can make educated guesses about the ratio of rock to ice present. Titania's density, approximately \( 1310 \, \text{kg/m}^3 \), suggests a significant presence of ice, meaning there are more icy compounds compared to rocky materials.

Understanding the density of celestial bodies can also impact other area studies, such as their potential for hosting life or being suitable for missions. This characteristic is pivotal when considering the geological and atmospheric sciences of moons, planets, and other objects beyond Earth.