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Bohr Orbits of a Satellite. A 20.0 -kg satellite circles the earth once every 2.00 \(\mathrm{h}\) in an orbit having a radius of 8060 \(\mathrm{km} .\) (a) Assuming that Bohr's angular-momentum result \((L=n h / 2 \pi)\) applies to satellites just as it does to an electron in the hydrogen atom, find the quantum number \(n\) of the orbit of the satellite. (b) Show from Bohr's angular momentum result and Newton's law of gravitation that the radius of an earth-satellite orbit is directly proportional to the square of the quantum number, \(r=k n^{2},\) where \(k\) is the constant of proportionality. (c) Using the result from part \((\mathrm{b}),\) find the distance between the orbit of the satellite in this problem and its next "allowed" orbit. (Calculate a numerical value.) (d) Comment on the possibility of observing the separation of the two adjacent orbits. (e) Do quantized and classical orbits correspond for this satellite? Which is the "correct" method for calculating the orbits?

Step by step solution

01

Calculate Orbital Angular Velocity

Determine the angular velocity \( \omega \) of the satellite using the formula \( \omega = \frac{2\pi}{T} \), where \( T \) is the period of orbit in seconds. Convert 2.00 hours to seconds: \( T = 2 \times 3600 = 7200 \) seconds. Thus, \( \omega = \frac{2\pi}{7200} \approx 8.73 \times 10^{-4} \) rad/s.

02

Calculate Angular Momentum

The angular momentum \( L \) is given by \( L = mvr = m \cdot \omega \cdot r^2 \) for a satellite. Plug in values: \( m = 20.0 \) kg, \( r = 8060 \times 10^3 \) m, \( \omega = 8.73 \times 10^{-4} \) rad/s.\[ L = 20.0 \times (8.73 \times 10^{-4}) \times (8060 \times 10^3)^2 \approx 1.137 \times 10^{16} \text{ J\cdot s}\].

03

Find Quantum Number n

Apply Bohr's quantization formula \( L = n \frac{h}{2\pi} \). With \( h = 6.626 \times 10^{-34} \) J·s, solve for \( n \):\[ n = \frac{L \times 2\pi}{h} \approx \frac{1.137 \times 10^{16} \times 2\pi}{6.626 \times 10^{-34}} \approx 1.08 \times 10^{34}\].

04

Prove Radius Proportionality Relation

Using Bohr's equation \( L = n \frac{h}{2\pi} \) and the centripetal force equation \( G\frac{Mm}{r^2} = \frac{m\omega^2 r}{r} \), equate \( \G\frac{Mm}{r} = mv^2 = m\left(\frac{L}{mr}\right)^2 \ \) and simplify to \( r = k n^2 \). Identify \( k = \frac{h^2}{4\pi^2GM} \).

05

Calculate Distance to Next Orbit

Use the proportional relation \( r = kn^2 \) to find the next allowed orbit \( r_{n+1} = k(n+1)^2 \). Calculate the difference \( \Delta r = k\left((n+1)^2 - n^2\right) \approx k(2n + 1) \). Thus \( \Delta r = 2kn \).

06

Determine Feasibility of Observation

The calculated orbit separation \( \Delta r \) is astronomically large, on the scale of kilometers to larger astronomical units. Such separations are not observable in practice for macroscale objects like satellites due to gravitational forces dominating over quantum effects.

07

Discuss Classical vs. Quantized Orbits

In large systems like satellites and planets, classical mechanics provides accurate calculations and predictions. Quantum effects, significant only at atomic scales, become irrelevant. Thus, classical treatment is the correct and practical method for satellite orbit calculations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the smallest scales. It is crucial for understanding phenomena that cannot be explained using classical mechanics, especially at atomic and subatomic levels. This theory introduces concepts such as quantization, wave-particle duality, and uncertainty.

In the Bohr Model, a key part of quantum mechanics, electrons orbit the nucleus in quantized orbits, meaning only certain discrete energy levels are allowed. This is similar to how Bohr's quantum condition is applied to predict satellite orbits in the exercise, which is, however, more theoretical for macroscopic objects like satellites.

• Relies on discrete energy levels and quantization.
• Only applicable primarily to atomic and molecular scales.
• Predicts phenomena that classical mechanics cannot explain.

While quantum mechanics accurately predicts the behavior of particles at small scales, at larger, macroscopic scales like a satellite, gravitational interactions described by classical physics become more significant.

Satellite Orbits

Satellite orbits refer to the path that satellites follow as they travel around celestial bodies, like Earth. These orbits are dictated by the gravitational forces acting between the satellite and the planet.

Orbits can be circular or elliptical, depending on the energy and velocity of the satellite. The exercise demonstrates calculating the orbital dynamics of a satellite using classical mechanics combined with the Bohr model to introduce quantum concepts.

• Circular or elliptical in shape.
• Determined by velocity and gravitational pull.
• Used for telecommunications, weather analysis, and scientific exploration.

Understanding satellite orbits is crucial for effective satellite deployment and optimizing space missions. In practice, orbits are calculated using classical mechanics, while the quantum approach provides a conceptual similarity to atomic models, not typically applied in practice.

Angular Momentum

Angular momentum is a measure of the rotational motion of an object. It is a key concept in both classical and quantum mechanics, conserved in isolated systems.

For a satellite in orbit, the angular momentum depends on its mass, velocity, and radius of orbit, calculated using the formula: \[ L = mvr = m \cdot \omega \cdot r^2 \]where \(L\) is angular momentum, \(m\) is mass, \(v\) is velocity, and \(r\) is radius.

• Conserved in isolated systems without external torques.
• Dependence on mass, velocity, and distance from the axis of rotation.
• Quantized in quantum systems, described by discrete values.

In the exercise, Bohr's quantization condition \( L = n \frac{h}{2\pi} \) applies the concept of discrete angular momentum levels, echoing its quantum origins but exploring its use in celestial mechanics.

Gravitational Force

Gravitational force is the attraction between two masses. It is a fundamental force in nature described by Newton's law of universal gravitation.

This force governs the motion of celestial bodies and is crucial for calculating satellite orbits. The gravitational force between two masses \( M \) and \( m \) separated by a distance \( r \) is given by:\[ F = G \frac{Mm}{r^2} \]where \(G\) is the gravitational constant.

• Acts between any two pieces of matter in the universe.
• Proportional to the product of the two masses.
• Inversely proportional to the square of the distance between them.

In orbit calculations, gravitational forces are balanced by the centripetal force necessary for maintaining the orbit. Thus, while quantum mechanics offers insight into small-scale interactions, gravitational force remains dominant in macroscopic scenarios like satellites.