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Chapter 12: Problem 21

Satellites in low orbit around the Earth lose energy from colliding with the gases of the upper atmosphere, causing them to slowly spiral inward. What happens to their kinetic energy as they fall inward?

Short Answer

Expert verified
Answer: As the satellite loses energy and spirals inward, its kinetic energy increases to maintain the conservation of mechanical energy. The satellite gains tangential velocity as it moves into an orbit closer to the Earth, resulting in the increase in kinetic energy.

Step by step solution

01

Understand gravitational potential energy

Gravitational potential energy (U) is the energy an object possesses due to its position relative to a massive body, such as Earth. It is given by the formula: U = -G * m * M / r where G is the gravitational constant, m is the mass of the satellite, M is the mass of the Earth, and r is the distance between the center of the Earth and the satellite.
02

Understand kinetic energy

Kinetic energy (K) is the energy of an object due to its motion. It can be calculated using the formula: K = (1/2) * m * v^2 where m is the mass of the satellite, and v is its velocity.
03

Understand conservation of mechanical energy

The total mechanical energy (E) of a system is the sum of its kinetic energy (K) and potential energy (U) - that is, E = K + U. According to the conservation of mechanical energy principle, the total mechanical energy of a system remains constant if no external force acts upon it. In this case, the external force is the collisions with the atmosphere. We'll need to calculate the changes in total mechanical energy due to these collisions.
04

Calculate the change in gravitational potential energy

As the satellite spirals inward, its distance (r) from the Earth's center decreases. This means that the gravitational potential energy of the satellite (U) becomes less negative, or in other words, it increases. Since the mass of the satellite and Earth and the gravitational constant do not change, the change in U is directly related to the change in r.
05

Calculate the change in kinetic energy

In order for the total mechanical energy to be conserved, the increase in gravitational potential energy must be matched by an increase in the satellite's kinetic energy. To maintain a stable orbit, as the satellite moves inward, its tangential velocity must increase. Using the formula K = (1/2) * m * v^2, the increase in kinetic energy can be calculated.
06

Summarize the changes in the satellite's kinetic energy

As the satellite loses energy due to collisions with the Earth's upper atmosphere, it spirals inward, leading to an increase in its gravitational potential energy. To conserve mechanical energy, the satellite's kinetic energy also increases, as it gains tangential velocity while moving into an orbit closer to the Earth. Thus, the satellite's kinetic energy increases as it falls inward.

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Most popular questions from this chapter

The distances from the Sun at perihelion and aphelion for Pluto are \(4410 \cdot 10^{6} \mathrm{~km}\) and \(7360 \cdot 10^{6} \mathrm{~km},\) respectively. What is the ratio of Pluto's orbital speed around the Sun at perihelion to that at aphelion?

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Newton was holding an apple of mass \(100 . \mathrm{g}\) and thinking about the gravitational forces exerted on the apple by himself and by the Sun. Calculate the magnitude of the gravitational force acting on the apple due to (a) Newton, (b) the Sun, and (c) the Earth, assuming that the distance from the apple to Newton's center of mass is \(50.0 \mathrm{~cm}\) and Newton's mass is \(80.0 \mathrm{~kg}\).

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