Problem 29 - College Physics

Question

Huygens probe on Titan. In January 2005 the Huygens probe landed on Saturn's moon Titan, the only satellite in the solar system having a thick atmosphere. Titan's diameter is \(5150 \mathrm{km},\) and its mass is \(1.35 \times 10^{23} \mathrm{kg}\) . The probe weighed 3120 \(\mathrm{N}\) on the earth. What did it weigh on the surface of Titan?

Step-by-Step Solution

Answer

The probe weighed approximately 430.3 N on the surface of Titan.

Step 1: Determine Gravitational Acceleration on Earth

The weight of an object is given by the formula \( W = m imes g \), where \( W \) is the weight, \( m \) is the mass, and \( g \) is the gravitational acceleration. On Earth, \( g = 9.8 \, \text{m/s}^2 \). To find the mass of the probe, use the weight on Earth: \( 3120 \, \text{N} = m \times 9.8 \, \text{m/s}^2 \). Solve for \( m \): \( m = \frac{3120}{9.8} \approx 318.37 \, \text{kg} \).

Step 2: Calculate Gravitational Acceleration on Titan

The gravitational acceleration \( g_t \) on Titan can be calculated using the formula \( g_t = \frac{G imes M}{R^2} \), where \( G \) is the universal gravitational constant \( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \), \( M \) is the mass of Titan, and \( R \) is the radius of Titan. First, convert the diameter of Titan to radius: \( R = \frac{5150}{2} = 2575 \, \text{km} = 2.575 \times 10^6 \, \text{m} \). Then substitute the values: \( g_t = \frac{6.674 \times 10^{-11} \, \times 1.35 \times 10^{23}}{(2.575 \times 10^6)^2} \approx 1.352 \, \text{m/s}^2 \).

Step 3: Determine the Weight of the Probe on Titan

We now use the mass of the probe and the gravitational acceleration on Titan to determine the weight of the probe: \( W_t = m \times g_t = 318.37 \, \text{kg} \times 1.352 \, \text{m/s}^2 \approx 430.3 \, \text{N} \). This is the weight of the Huygens probe on Titan's surface.

Key Concepts

Gravitational acceleration

Gravitational acceleration is the rate at which an object accelerates due to the force of gravity. On Earth, this value is approximately 9.8 \, \text{m/s}^2. Each planet or celestial body has its own gravitational acceleration, which depends on its mass and size.

The gravitational force between two masses, such as an object and a planet, is calculated using the formula: \[ F = \frac{G \times M \times m}{R^2} \]Where:

• \(G\) is the universal gravitational constant \(6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\)
• \(M\) is the planet's mass
• \(m\) is the mass of the object
• \(R\) is the distance between the centers of the two masses (radius of the planet if the object is on the surface)

By measuring the gravitational acceleration, we can determine the weight of any object on a planet, which varies because gravitational forces in space differ from those we feel on Earth.
Understanding gravitational acceleration is crucial for calculating how much force an object experiences and is fundamental to planetary science.

Huygens probe

The Huygens probe was a part of the Cassini-Huygens mission, representing a significant milestone in space exploration. It was designed to explore the atmosphere and surface of Titan, Saturn's largest moon, which is unique due to its thick atmosphere.

Launched in 1997 by NASA, ESA, and ASI, the probe provided invaluable data upon its landing on Titan in 2005. Key contributions of the Huygens probe:

• First and only landing on Titan, providing images and atmospheric data
• Revealed Titan's surface features, including lakes and rivers made of hydrocarbons
• Helped confirm the presence of complex organic chemistry

The Huygens probe's successful landing expanded our understanding of celestial bodies with thick atmospheres and laid the groundwork for future planetary exploration missions.

Weight on different planets

The weight of an object is not constant across different planets or celestial bodies. Weight is the force with which gravity pulls an object toward the surface of a planet. It is calculated by the formula:\[ W = m \times g\]Where:

• \(W\) is weight
• \(m\) is mass
• \(g\) is the gravitational acceleration specific to the location

While mass remains the same, gravitational acceleration \(g\) varies depending on the mass and radius of the planet being considered. For example, the Huygens probe weighed 3120 N on Earth (with \(g \approx 9.8 \, \text{m/s}^2\)), but on Titan, where \(g_t \approx 1.352 \, \text{m/s}^2\), it weighed approximately 430 N.
Understanding weight variations across planets is crucial for space missions, as it impacts how equipment is designed and how astronauts would move and perform tasks in different gravitational environments.

Planetary science

Planetary science is the study of planets, moons, and planetary systems, mainly within our solar system. It overlaps with many areas of physics, chemistry, and geology. The field involves:

• Studying planet atmospheres, surfaces, and interiors
• Understanding the formation and evolution of solar systems
• Comparing Earth to other planets to find potential signs of life

Tools in planetary science include telescopes, spacecraft, and landers like the Huygens probe, which gather data directly from bodies like Titan.
By examining both the Earth and other celestial bodies, planetary science seeks to answer profound questions about our origins, the potential for life elsewhere, and the fundamental processes that have shaped our solar system.