Given that
Mass of satellite m = 10 kg
Radius of orbit r = 8000 km = 8 * 10^{6} m
Time period T = 2 h = 7200 seconds
Assuming that the angular momentum postulate of Bohr applies to the satellite
Step 1: Using the Bohr's postulate to get an expression for quantum number
The second postulate of Bohr describes certain stable orbits. This postulate states that the electron revolves around the nucleus only in certain orbits where the angular momentum is some integral multiple of \frac{h}{2 \pi} .
L = m v_n r_n = \frac{nh}{2 \pi} ..................(1)
Relation between the angular and linear velocity
Velocity v_n = \omega r_n = \frac{2 \pi r_n}{T}
Substituting v_n . value in equation (1)
[math] m [\frac{2 \pi r_n}{T}] r_n = \frac{nh}{2 \pi} [/math]
n = m \frac{4 \pi^2 r_n^2}{T * h}
Thus, Quantum number of the satellite orbit n = m \frac{4 \pi^2 r_n^2}{T * h}
Step 2: Substitute the known values in the above equation
n = 10 \frac{4 * (3.14)^2 * (8 * 10^{6})^2 }{7200 * 6.63 * 10^{-34}}
n = 5.3 * 10^{45}
Hence, Quantum number of the satellite orbit n = 5.3 * 10^{45}
The quantum number of the motion of the satellite is extremely high! In reality, the effects of quantization conditions are more likely to be classical physics for such large numbers.
