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The Ottoman Empire

On a hypothet...

Question

On a hypothetical planet, satellite can only revolve in quantized energy level, i.e., magnitude of energy of a satellite is integer multiple of a fixed energy. If two successive orbits have radius $R$ and $\frac{3 R}{2}$ , what could be maximum radius of the satellite in terms of $R$ ?

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Solution

Formula used:
$K E = \frac{G M m}{2 r}$

For energy in radius
$r_{i} = ∣ ∣ ∣ \frac{- G M m}{2 r} ∣ ∣ ∣ = n k$

where, $n$ is integer and $k$ is constant. Now, according to the given situation,

For $r = R$

$\frac{G M m}{2 R} = n k \dots . . (i)$

For energy in radius

$r_{f} = ∣ ∣ ∣ \frac{- G M m}{2 r} ∣ ∣ ∣ = (n - 1) k$

According to the given situation,
For $r = \frac{3 R}{2}$

$\frac{G M m}{2 (\frac{3 R}{2})} = (n - 1) k \dots . . (i i)$

Solving $(i)$ and $(i i)$ , we get

$k = \frac{G M m}{6 R}$

Now this implies that for $R_{m a x}$

$\frac{G M m}{2 R_{m a x}} = (1) \frac{G M m}{6 R}$

$R_{m a x} = 3 R$

Final Answer: $3$

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On a hypothetical planet, satellite can only revolve in quantized energy level, i.e., magnitude of energy of a satellite is integer multiple of a fixed energy. If two successive orbits have radius R and 3R2, what could be maximum radius of the satellite in terms of R?

On a hypothetical planet, satellite can only revolve in quantized energy level, i.e., magnitude of energy of a satellite is integer multiple of a fixed energy. If two successive orbits have radius $R$ and $\frac{3 R}{2}$ , what could be maximum radius of the satellite in terms of $R$ ?