Question

After a spacewalk, a 1.00 -kg tool is left \(50.0 \mathrm{~m}\) from the center of gravity of a 20.0 -metric ton space station, orbiting along with it. How much closer to the space station will the tool drift in an hour due to the gravitational attraction of the space station?

Short answer

Answer: The tool drifts approximately 17.3 μm closer to the space station in an hour due to the gravitational attraction between them.

Step-by-step solution

Identify the given values

We are given: - Mass of the tool, \(m_1 = 1.00 \hspace{2mm} kg\) - Mass of the space station, \(m_2 = 20.0 \hspace{2mm} metric \hspace{2mm} tons = 20,000 \hspace{2mm} kg\) - Initial distance between the center of mass of the tool and space station, \(r = 50.0 \hspace{2mm} m\) - Time elapsed, \(t = 1 \hspace{2mm} hour = 3600 \hspace{2mm} seconds\)

Write down the Law of Universal Gravitation formula

The gravitational force between two objects with masses \(m_1\) and \(m_2\), separated by a distance \(r\), is given by the formula: \(F = G\frac{m_1 \cdot m_2}{r^2}\) where \(G\) is the gravitational constant, \(G = 6.674 \times 10^{-11} \hspace{2mm} N\cdot m^2/kg^2\).

Calculate the gravitational force between the tool and the space station

Using the given values, we can calculate the gravitational force between the tool and the space station: \(F = (6.674 \times 10^{-11})\frac{1.00 \cdot 20,000}{50.0^2}\) \(F = 2.6696 \times 10^{-9} \hspace{2mm} N\)

Calculate the acceleration of the tool towards the space station

Using Newton's second law of motion, \(F = m \cdot a\), we can find the acceleration of the tool towards the space station: \(a = \frac{F}{m_1}\) \(a = \frac{2.6696 \times 10^{-9}}{1.00}\) \(a = 2.6696 \times 10^{-9} \hspace{2mm} m/s^2\)

Calculate the distance the tool drifts towards the space station in an hour

Using the equation of motion, \(d = \frac{1}{2} a \cdot t^2\), we can calculate the distance the tool drifts towards the space station in an hour: \(d = \frac{1}{2} (2.6696 \times 10^{-9})(3600)^2\) \(d \approx 17.3 \times 10^{-6} \hspace{2mm} m\) The tool drifts approximately \(17.3 \mathrm{\mu m}\) closer to the space station in an hour due to the gravitational attraction between them.

Key concepts

Law of Universal Gravitation

Understanding the Law of Universal Gravitation is essential in predicting how celestial objects interact with each other due to the force of gravity. According to this law, formulated by Sir Isaac Newton, every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

This concept is expressed mathematically as: \[\begin{equation}F = G \frac{m_1 \cdot m_2}{r^2}\end{equation}\]where:

• \( F \) is the gravitational force between the two masses;
• \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} N \cdot m^2/kg^2 \));
• \( m_1 \) and \( m_2 \) are the masses of the two objects;
• \( r \) is the distance between the centers of the masses.

The significance of this law is profound, as it explains not only the attraction between objects on Earth but also the motion of planets, stars, and galaxies in the universe.

Newton's Second Law of Motion

Newton's second law of motion provides the relationship between an object's mass, the net force acting upon it, and its acceleration. The net force applied to an object causes it to accelerate in the direction of the force. This is expressed by the equation:\[\begin{equation}F = m \cdot a\end{equation}\]where:

• \( F \) stands for the force exerted on the object;
• \( m \) represents the mass of the object;
• \( a \) is the acceleration experienced by the object.

This law informs us that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Thus, for a given force, a more massive object accelerates less than a less massive one. In the context of gravitational attraction, this law allows us to understand how an object, like a tool in space, will accelerate towards a larger mass, such as a space station, when a gravitational force is applied.

Equation of Motion

The equation of motion is a mathematical expression used to calculate the displacement of an object when its acceleration (a) and the time of motion (t) are known. The simplest form, assuming the initial velocity is zero and the acceleration is constant, is given by:\[\begin{equation}d = \frac{1}{2} a \cdot t^2\end{equation}\]where:

• \( d \) is the displacement;
• \( a \) is the constant acceleration;
• \( t \) is the time elapsed.

This formula is particularly useful in space scenarios where external forces, except for gravity, are minimal, and the acceleration can be considered constant. By combining this equation with the Law of Universal Gravitation and Newton's second law of motion, we are capable of predicting how far objects will move towards each other due to gravitational forces over a given period, as seen in the exercise example with the drifting tool and space station.