Titania, the largest moon of the planet Uranus, has 1/8 the radius of the earth and 1/1700 the mass of the earth. What is the average density of Titania? (This is less than the density of rock, which is one piece of evidence that Titania is made primarily of ice.)

검증된 단계별 안내

Start by recalling the formula for density, which is defined as mass divided by volume: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).

To find the volume of Titania, use the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \). Since Titania has 1/8 the radius of Earth, its radius \( r_T \) is \( r_T = \frac{1}{8} r_E \), where \( r_E \) is the radius of Earth.

Substitute \( r_T = \frac{1}{8} r_E \) into the volume formula: \( V_T = \frac{4}{3} \pi \left( \frac{1}{8} r_E \right)^3 \). Simplify this expression to find the volume of Titania in terms of Earth's volume.

Given that Titania has 1/1700 the mass of Earth, express Titania's mass \( M_T \) as \( M_T = \frac{1}{1700} M_E \), where \( M_E \) is the mass of Earth.

Finally, substitute the expressions for \( M_T \) and \( V_T \) into the density formula: \( \text{Density of Titania} = \frac{\frac{1}{1700} M_E}{\frac{4}{3} \pi \left( \frac{1}{8} r_E \right)^3} \). Simplify this expression to find the average density of Titania in terms of Earth's mass and radius.

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주요 개념

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Density

Density is a measure of mass per unit volume, typically expressed in kilograms per cubic meter (kg/m³). It is calculated by dividing the mass of an object by its volume. Understanding density is crucial for determining the composition of celestial bodies, as it provides insights into their internal structure and material makeup.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius of the sphere. This formula is essential for determining the volume of spherical objects like moons and planets, which is a necessary step in calculating their density when the mass is known.

Gravitational Scaling

Gravitational scaling involves understanding how changes in size and mass affect gravitational properties. For celestial bodies, knowing how the radius and mass compare to Earth helps in calculating other properties like density. In this context, Titania's radius and mass relative to Earth are used to find its volume and subsequently its density.