Problem 52 - University Physics with Modern Physics

Question

A landing craft with mass 12,500 kg is in a circular orbit 5.75 \(\times\) 10\(^5\) m above the surface of a planet. The period of the orbit is 5800 s. The astronauts in the lander measure the diameter of the planet to be 9.60 \(\times\) 10\(^6\) m. The lander sets down at the north pole of the planet. What is the weight of an 85.6-kg astronaut as he steps out onto the planet's surface?

Step-by-Step Solution

Answer

The weight of the astronaut on the surface is the gravitational force calculated in Step 4.

Step 1: Calculate the Radius of the Planet

The diameter of the planet is given as \(9.60 \times 10^6\) meters. To find the radius, divide the diameter by 2. \[R_p = \frac{9.60 \times 10^6}{2} = 4.80 \times 10^6 \text{ m}\]

Step 2: Calculate the Radius of the Orbit

The orbit's altitude is given as \(5.75 \times 10^5\) meters above the planet's surface. Therefore, the total radius of the orbit \(R_o\) is the sum of the planet's radius and the altitude above it. \[R_o = R_p + 5.75 \times 10^5 = 4.80 \times 10^6 + 5.75 \times 10^5 = 5.375 \times 10^6 \text{ m}\]

Step 3: Use Orbital Mechanics to Find Planet's Mass

The formula for the period \(T\) of a circular orbit is:\[T = 2\pi \sqrt{\frac{R_o^3}{G M_p}}\]where \(M_p\) is the mass of the planet and \(G\) is the gravitational constant \(6.674 \times 10^{-11}\, \text{Nm}^2\text{kg}^{-2}\). Rearrange to solve for \(M_p\):\[M_p = \frac{4\pi^2 R_o^3}{G T^2}\]Now plug in the known values:\[M_p = \frac{4\pi^2 (5.375 \times 10^6)^3}{6.674 \times 10^{-11} \times (5800)^2}\]

Step 4: Calculate Astronaut's Weight on the Surface

Once the mass of the planet \(M_p\) is found, use the formula for gravitational force: \[F = \frac{G M_p m}{R_p^2}\]where \(m = 85.6\, \text{kg}\) is the mass of the astronaut. Plug in the values to find the weight:\[F = \frac{6.674 \times 10^{-11} \times M_p \times 85.6}{(4.80 \times 10^6)^2}\]

Key Concepts

Circular Orbit

A circular orbit refers to the path that an object follows when it moves around a celestial body, such as a planet, in a perfect circle. An object's orbit is considered circular when the distance from the center of the celestial body remains constant. In this context, the landing craft moves in a circular orbit around the planet, maintaining a steady altitude of 5.75 \( \ imes \ 10^5 \) meters above the planet's surface.
The key aspects of a circular orbit include:

• Constant Speed: In a circular orbit, the moving craft maintains a consistent speed as it revolves around the planet.
• Gravitational Force: The gravitational pull of the planet acts as the centripetal force that keeps the craft in its circular path.
• Period of Orbit: This is the time it takes for the craft to complete one full circle around the planet. In this exercise, it is given as 5800 seconds.

Understanding circular orbits is crucial in orbital mechanics as it helps in predicting the movement and behavior of satellites and spacecraft.

Planetary Mass

The planetary mass is a fundamental property that influences the gravitational force experienced by objects within the vicinity of a planet. It determines how strongly an object is attracted to the planet's surface.
To find the mass of the planet in the exercise, we make use of the formula related to circular orbits. By rearranging the formula for the orbital period:\[ T = 2\pi \sqrt{\frac{R_o^3}{G M_p}} \]We solve for the planet's mass \( M_p \), which becomes:\[ M_p = \frac{4\pi^2 R_o^3}{G T^2} \]In this equation:

• \( T \) is the orbital period, given as 5800 s.
• \( R_o \) is the radius of the entire orbit, calculated from the planet's radius plus the orbit's height.
• \( G \) is the universal gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2} \).

By substituting these known values into the formula, we can determine the planetary mass which affects how the gravitational force is calculated.

Orbital Mechanics

Orbital mechanics is the study of the motions of artificial and natural celestial bodies under the influence of forces like gravity. It combines principles from physics and mathematics to explain how objects move through space.
In this exercise, the focus is on the application of orbital mechanics to determine the weight of an astronaut as they step onto the planet's surface. After calculating the planet's mass using the orbital period, we apply the gravitational force equation:\[ F = \frac{G M_p m}{R_p^2} \]This helps us find the force (or weight) acting on the astronaut:

• \( G \) is the gravitational constant.
• \( M_p \) is the mass of the planet derived earlier.
• \( m \) is the astronaut's mass, which is 85.6 kg.
• \( R_p \) is the planet's radius.

The exerted force \( F \) is what we perceive as weight, illustrating the practical application of orbital mechanics to solve real-world problems related to space exploration.