Question

One of the Echo satellites consisted of an inflated aluminum balloon $30\mathrm{m}$ in diameter and of mass $20\mathrm{kg}$. A meteor having a mass of $7.0\mathrm{kg}$ passes within $3.0\mathrm{m}$ of the surface of the satellite. If the effect of all bodies other than the meteor and satellite are ignored, what gravitational force does the meteor experience at closest approach to the satellite?

Short answer

The gravitational force that the meteor experiences at closest approach to the satellite is $F=1.04\times {10}^{-9}N$.

Step-by-step solution

Identify the known variables.

We know that the mass of the satellite (m1) is 20 kg, the mass of the meteor (m2) is 7.0 kg, the diameter of the satellite d is 30 m, therefore the radius (r1) is 15 m, and the distance of closest approach (d) is 3.0 m.

Calculate the total distance between centers of the meteor and the satellite (r).

Since the meteor passes with 3 m from the surface of the satellite, the total distance (r) between their centers is the sum of the satellite radius (r1) and the closest approach (d). So, $r=r1+d=15m+3m=18m$.

Calculate the gravitational force (F).

Now we can substitute the known values into the universal gravitational force formula. The gravitational force (F) then is: $F=G\cdot (m1\cdot m2)/r^2=6.674\times {10}^{-11}N\cdot m^2/k{g}^2\cdot (20kg\cdot 7kg)/(18m{)}^2$. After making the calculation we find that $F=1.04\times {10}^{-9}N$.

Key concepts

Universal Gravitational Constant

In physics, the Universal Gravitational Constant, often symbolized as **G**, plays a critical role in calculating the gravitational force between two objects. This constant is a fundamental piece of Newton's Law of Universal Gravitation, which describes how the force of gravity acts.
The value of the Universal Gravitational Constant is approximately **6.674 × 10^-11 N·m²/kg²**. This number is quite small, indicating that gravity is a very weak force compared to other fundamental forces. Yet, it is remarkably significant due to its omnipresent influence, structuring entire galaxies and impacting the dynamics of celestial bodies.
In the given exercise, this constant is crucial to compute the gravitational force that the Echo satellite exerts on a passing meteor. By applying this constant to the formula:
$$\text{Math input error}$$
where **m₁** and **m₂** are the masses of the two objects and **r** is the distance between their centers, we can determine the force involved.

Mass of celestial bodies

The mass of celestial bodies significantly impacts the gravitational forces they exert. In space, objects like planets, stars, and satellites have vast masses, leading to substantial gravitational effects.
In this exercise, you encounter two celestial masses:

• The Echo satellite, with a mass of 20 kg
• A meteor, weighing 7 kg

Understanding these masses is essential for solving gravitational problems, as they directly affect the resulting force. Larger masses increase the gravitational pull, making understanding this factor critical in physics.
Masses are typically denoted by **m** in equations. Here, they are plugged into the gravitational formula to calculate the force of attraction, highlighting mass as a core component.

Distance between objects

The distance between two objects is crucial in determining the strength of the gravitational force between them. Gravity diminishes rapidly with distance, following an inverse square law.
In practice, this means that the closer two objects are, the stronger the gravitational pull between them. The formula
$$F=G\frac{m_1\times m_2}{r^2}$$
illustrates this relationship, where **r** represents the distance between the centers of the two bodies. The force is inversely proportional to the square of this distance, meaning that if the distance doubles, the force is reduced to one-quarter of its original value.
In the exercise above, the challenge involves a meteor passing close to an Echo satellite. The distance from the meteor to the satellite's surface is 3 meters, but to find the correct **r** value for our equation, we sum this with the satellite's radius. Thus, the total distance **r** is 18 meters, affecting the gravity interaction significantly.

Physics problem-solving

Physics problem-solving requires a structured approach to tackling complex questions. This ensures accurate results and strengthens your understanding of fundamental concepts.
When solving for gravitational force, begin by identifying known variables:

• Mass of each object
• Distance between their centers
• Any constants required (like **G** in gravitational problems)

Next, insert these variables into the appropriate formula. Ensure all measurements are in compatible units, often the SI units, such as kilograms for mass and meters for distance.
Calculate step-by-step, checking your work for accuracy. In our exercise, the formula used combined masses and distance:
$$F=G\frac{m_1\times m_2}{r^2}$$
After performing these calculations, you get the gravitational force acting between the meteor and the Echo satellite. Such a methodical problem-solving method with attention to detail and logic is key to mastering physics challenges.