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(D) The magnitude of the gravitational potential energy of a satellite in an orbit of radius $r$ is given by $E = \frac{GMm}{2r}$.According to the pro

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$3$

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(D) The magnitude of the gravitational potential energy of a satellite in an orbit of radius $r$ is given by $E = \frac{GMm}{2r}$.According to the problem,the energy is quantized such that $E_n = n \cdot k$,where $n$ is an integer and $k$ is a constant energy.For the first orbit with radius $R$,we have $\frac{GMm}{2R} = n \cdot k$.For the next successive orbit with radius $\frac{3R}{2}$,the energy must be $(n-1) \cdot k$ (since energy magnitude decreases as radius increases),so $\frac{GMm}{2(3R/2)} = (n-1) \cdot k$,which simplifies to $\frac{GMm}{3R} = (n-1) \cdot k$.Dividing the two equations: $\frac{GMm/2R}{GMm/3R} = \frac{n}{n-1} \Rightarrow \frac{3}{2} = \frac{n}{n-1}$.Solving for $n$: $3n - 3 = 2n \Rightarrow n = 3$.Substituting $n=3$ back into the first equation: $\frac{GMm}{2R} = 3k \Rightarrow k = \frac{GMm}{6R}$.The maximum radius $R_{	ext{max}}$ corresponds to the lowest energy level,which is $n=1$.Thus,$\frac{GMm}{2R_{	ext{max}}} = 1 \cdot k = \frac{GMm}{6R}$.Solving for $R_{	ext{max}}$ gives $R_{	ext{max}} = 3R$.

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

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