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The problem you have provided involves the calculation of the orbital parameters for a satellite in circular orbit around Jupiter. Let me break it down:

### Part a: Newton’s Laws and Orbital Equations

Given the mass of Jupiter \( M_J = 1.90 \times 10^{27} \, \text{kg} \) and its radius \( R_J = 7.14 \times 10^7 \, \text{m} \), we are to derive two results using Newton's laws:

1. **Orbital speed \( v \)**:
   
   The gravitational force provides the necessary centripetal force for a satellite in orbit. Thus, by equating gravitational force to centripetal force:
   
   \[
   \frac{GM_J}{R^2} = \frac{v^2}{R}
   \]
   where \( G \) is the gravitational constant, \( M_J \) is the mass of Jupiter, and \( R \) is the radius of the orbit.

   Solving for \( v \):
   
   \[
   v = \sqrt{\frac{GM_J}{R}}
   \]

   This is the formula for the orbital speed.

2. **Orbital period \( T \)**:
   
   The orbital period \( T \) can be found from the relation:
   
   \[
   T = \frac{2\pi R}{v}
   \]
   
   Substituting \( v = \sqrt{\frac{GM_J}{R}} \) into this equation:
   
   \[
   T = 2\pi \sqrt{\frac{R^3}{GM_J}}
   \]

   This matches the form \( T = \sqrt{\frac{4\pi^2 R^3}{GM_J}} \).

### Part b: Synchronization with Jupiter’s Rotation

Here, we are tasked with synchronizing the satellite’s orbit with Jupiter’s rotation period. The rotational period of Jupiter is given as \( T_J = 3.55 \times 10^4 \, \text{s} \).

Using the formula for the period of the orbit derived earlier:

\[
T = 2\pi \sqrt{\frac{R^3}{GM_J}}
\]

We are solving for the orbital radius \( R \), given that the period of the satellite’s orbit should equal Jupiter’s rotational period \( T_J \). Thus:

\[
T_J = 2\pi \sqrt{\frac{R^3}{GM_J}}
\]

Squaring both sides:

\[
T_J^2 = 4\pi^2 \frac{R^3}{GM_J}
\]

Solving for \( R^3 \):

\[
R^3 = \frac{GM_J T_J^2}{4\pi^2}
\]

Finally, solving for \( R \):

\[
R = \left( \frac{GM_J T_J^2}{4\pi^2} \right)^{1/3}
\]

Substitute the values:

- \( G = 6.674 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)
- \( M_J = 1.90 \times 10^{27} \, \text{kg} \)
- \( T_J = 3.55 \times 10^4 \, \text{s} \)

Now, calculate \( R \).

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### 5 Related Questions:
1. How does the gravitational force affect a satellite’s orbit?
2. Why do we use Newton’s laws to derive orbital speed?
3. What is the significance of orbital synchronization with a planet's rotation?
4. How does the mass of the planet influence the orbital radius?
5. What happens to a satellite's orbit if its speed changes?

### Tip:
Always ensure units are consistent when calculating orbital dynamics to avoid errors in the final result.

An explorer plans a mission to place a satellite into a circular orbit around Jupiter. Calculate the orbital speed and period using Newton's laws, and find the orbital radius for synchronization with Jupiter's rotation period.

Learn how to calculate the orbital speed, period, and synchronization radius for a satellite orbiting Jupiter using Newton's laws. Includes detailed explanations of gravitational force, circular motion, and orbital mechanics.

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