Bohr Orbits of a Satellite. A 20.0 -kg satellite circles the...

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?

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?

Key Concepts

Observability of Quantized Orbits

While quantization is a fundamental aspect of microscopic systems, for macroscopic bodies such as satellites, the energy levels or orbital radii differences predicted by quantization are so minutely spaced that they cannot be experimentally resolved. This concept underscores the practical irrelevance of applying quantum quantization to large-scale systems, where classical mechanics remains the accurate and preferred method.

Quantum-Classical Correspondence

This principle deals with the way quantum mechanical results converge to classical predictions in the limit of large quantum numbers. For macroscopic objects like satellites, the quantum number becomes extremely high, meaning the spacing between quantized orbits is negligibly small, and classical orbital mechanics provides an adequate and more practical description of the motion.

Newtonian Orbital Mechanics

Newton’s law of gravitation and the concept of centripetal force underlie classical orbital mechanics. For a body in circular orbit, the gravitational force provides the necessary centripetal force. This balance determines the orbital speed and allows derivation of relationships between orbital radius, period, and mass. In this problem context, it is used alongside the quantization condition to derive relationships between the quantum number and the orbital radius.

Angular Momentum Quantization

This concept originates from Bohr’s model of the atom, where the angular momentum of an orbiting particle is restricted to discrete values given by L = n h/(2?). It means that only certain, quantized orbits are allowed, and it serves as a starting point for quantizing orbital systems. In the context of the problem, it involves applying the same angular momentum quantization postulate to a satellite orbiting the Earth.

Relationship Between Quantum Number and Orbital Radius

When Bohr’s quantization condition is combined with the classical equation for centripetal acceleration under gravity, the orbital radius is found to be proportional to the square of the quantum number (r ? n²). This is analogous to the hydrogen atom where orbital radii scale as n². It is a key derivation used to understand how quantized states would manifest even in gravitational systems.

Transcript

00:01 In this question, you're basically applying a boss model to a gravitational perspective.

00:10 Rather than an electron orbiting around a nucleus, we have a satellite orbiting around our earth.

00:20 Let's see where the boss model would work as well.

00:25 For this case, well, the only difference is that instead of kulom interactions, we have gravitational forces acting as well.

00:33 As the main source of centripetal acceleration.

00:39 So we start off this question using the boss postulate, that is the angular momentum, mvr, is now an integer multiple of h over 2 pi.

00:55 So n is an integer, an element of the positive integers.

01:04 We want to find what is n, which is equals to vr times 2 pi divided by h bar sorry h by h now a few things we do not have is the velocity so but we can convert velocity into period right since velocity times the period must give us the circumference of the entire orbit therefore the velocity is just 2 r over t so it's substituted into here we get 2 pi m times 2 pi r over t times r over h this gives us 4 pi square m r square over t times h so just i put all the values in so in mass of the satellite is given to be 20 kg 20 kg and the radius of the orbits is 8 .06 10 power 6.

02:29 The period is given to be in terms of 2 hours.

02:34 So i'll convert that 2 hours into seconds.

02:41 And h is the planks constant.

02:49 Therefore we should get the quantum number over here to be 1 .08 times 10 power 46.

03:02 For the next part we want to show that bore's angular momentum and the newton's gravitational law is directly proportional to the square of the quantum number.

03:20 Basically the radius is k times n square.

03:30 Now the way that we're going to do that is where we will need to use the circular motion equation, that is the acceleration, centritor acceleration, is v square over r.

03:43 This is equal to the gravitational acceleration due to gravity.

03:49 So this g times m e over r square.

03:55 Therefore v square is gm e over r.

04:03 Now since we have from our previous answer that are right from actually over here we can simplify this a bit...

Please give Ace some feedback