The magnitude of the tidal force between the International...

The magnitude of the tidal force between the International Space Station (ISS) and a nearby astronaut on a spacewalk is approximately 2$G m M a / r^{3} .$ In this expression, $M$ is the mass of the Earth, $r=6.79 \times 10^{6} \mathrm{m}$ is the distance from the center of the Earth to the orbit of the ISS, $m=125 \mathrm{kg}$ is the mass of the astronaut, and $a=10 \mathrm{m}$ is the distance from the astronaut to the center of mass of the ISS. (a) Calculate the magnitude of the tidal force for this astronaut. This force tends to separate the astronaut from the ISS if the astronaut is located along the line that connects the center of the Earth with the center of mass of the ISS. (b) Calculate the force of gravitational attraction between the astronaut and the ISS if they are 10 $\mathrm{m}$ apart and the ISS has a mass of $420,000 \mathrm{kg}$ . (c) Is the ISS orbit inside or outside the Roche limit?

Key Concepts

Roche Limit

The Roche limit is the minimum distance at which a celestial body, held together solely by its own gravity, can orbit a larger body without being torn apart by tidal forces. This concept is important for understanding whether the ISS is subject to significant tidal disruptions or remains intact, and it provides insight into the conditions under which gravitational forces from a large body overcome the self-gravity of an orbiting object.

Tidal Force

Tidal force arises due to the difference in gravitational pull experienced by an extended body in a non-uniform gravitational field. In this context, the term explains the gradient in the gravitational field of the Earth acting differently on separate parts of the astronaut and ISS system, causing them to experience a stretching force. This concept is essential in understanding how gravitational forces vary with distance and how they can lead to internal stresses or separation between nearby objects.

Gravitational Attraction

Gravitational attraction refers to the force that objects with mass exert on one another according to Newton’s law of universal gravitation. This concept is used in determining the direct attraction between two masses at a given separation distance, such as between the astronaut and the ISS. It allows for a comparison between the tidal forces experienced due to Earth's gravity and the mutual gravitational pull between the astronaut and the ISS.

Transcript

00:01 In this problem, we want to calculate the magnitude of the tidal force between the international space station and a nearby astronaut on a spacewalk.

00:09 The tidal force follows the equation f -little t equals 2 times big g, m, big m, a over r -quipped.

00:20 Here, big g is newton's gravitational constant.

00:23 M is the mass of the astronaut, big m is the mass of the earth, a is the distance between the astronaut to the center of mass of the iss, and r is the distance between the center of the earth to the iss.

00:39 So to find the magnitude of the tidal force, we go ahead and plug in all of these variables in the formula for title force.

00:49 Here we have title force will be equal to 2 times 6 .67 times 10 to the negative of 11, newton's meters squared over kilograms squared times mass of the astronaut, which is 125 kilograms, times mass of the space station, which is 4 .2 times 10 to the 5 kilograms, times the distance between the astronaut and the space station, which is 10 meters, all divided by the distance between the center of the earth to the iss, which here it is 6 .79 times 10 to the 6 meters.

01:54 And the denominator, the r is cubed.

01:58 And if we plug all that in into a calculator, we get about 3 .18 times 10 to the negative 3 newtons for the tidal force.

02:11 Now the second part of the problem wants us to find the gravitational force between the international space station and the astronaut.

02:20 So for this, remember that f of big g is equal to big g, m1, m2 divided by r square...

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