Consider a planet with mass mp, which has a moon with mass mm. Both objects are orbiting around a star of mass Ms. The position vectors from the star to the planet and its moon are rp and rm, respectively, as shown in Figure 1. Find the vector force on the moon mm due to gravity of the planet mp.

To find the vector force on the moon (mass \( m_m \)) due to the gravitational attraction of the planet (mass \( m_p \)), we will use Newton's law of gravitation, which gives the force between two masses.

### Newton’s Law of Gravitation:

\[
\mathbf{F} = - G \frac{m_p m_m}{r^2} \hat{r}
\]

Where:
- \( G \) is the gravitational constant.
- \( m_p \) is the mass of the planet.
- \( m_m \) is the mass of the moon.
- \( r \) is the distance between the planet and the moon.
- \( \hat{r} \) is the unit vector pointing from the planet to the moon.

### Step-by-Step Solution:

1. **Define the relative position vector:**
   Let \( \mathbf{r}_{pm} \) be the vector from the planet to the moon:
   \[
   \mathbf{r}_{pm} = \mathbf{r}_m - \mathbf{r}_p
   \]
   Where:
   - \( \mathbf{r}_m \) is the position vector of the moon relative to the star.
   - \( \mathbf{r}_p \) is the position vector of the planet relative to the star.

2. **Magnitude of the relative position vector:**
   The magnitude \( r \) of the relative position vector \( \mathbf{r}_{pm} \) is the distance between the planet and the moon:
   \[
   r = |\mathbf{r}_{pm}| = |\mathbf{r}_m - \mathbf{r}_p|
   \]

3. **Unit vector in the direction from the planet to the moon:**
   The unit vector \( \hat{r}_{pm} \) that points from the planet to the moon is given by:
   \[
   \hat{r}_{pm} = \frac{\mathbf{r}_{pm}}{|\mathbf{r}_{pm}|} = \frac{\mathbf{r}_m - \mathbf{r}_p}{|\mathbf{r}_m - \mathbf{r}_p|}
   \]

4. **Gravitational force on the moon due to the planet:**
   The gravitational force \( \mathbf{F}_{pm} \) on the moon due to the planet is:
   \[
   \mathbf{F}_{pm} = - G \frac{m_p m_m}{|\mathbf{r}_m - \mathbf{r}_p|^2} \frac{\mathbf{r}_m - \mathbf{r}_p}{|\mathbf{r}_m - \mathbf{r}_p|}
   \]
   Simplifying:
   \[
   \mathbf{F}_{pm} = - G \frac{m_p m_m}{|\mathbf{r}_m - \mathbf{r}_p|^3} (\mathbf{r}_m - \mathbf{r}_p)
   \]

### Final Expression for the Gravitational Force:

The vector force on the moon \( m_m \) due to the planet \( m_p \) is:
\[
\boxed{\mathbf{F}_{pm} = - G \frac{m_p m_m}{|\mathbf{r}_m - \mathbf{r}_p|^3} (\mathbf{r}_m - \mathbf{r}_p)}
\]

This is the gravitational force vector, which points from the moon towards the planet and has a magnitude proportional to the product of the two masses and inversely proportional to the square of the distance between them.

### Do you want any more details or clarifications?

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Here are some relative questions to deepen understanding:

1. How does the force change if the distance between the planet and moon doubles?
2. How would you express the gravitational force in terms of coordinates if the vectors \( \mathbf{r}_p \) and \( \mathbf{r}_m \) were given in a coordinate system?
3. What is the significance of the negative sign in the gravitational force equation?
4. How would the force on the moon change if the mass of the planet were doubled?
5. How does this equation relate to the force on the planet due to the moon?

### Tip:
Always pay attention to the direction of vectors in gravitational problems, as forces are vectors and their direction is crucial in solving such problems accurately.

Learn how to calculate the gravitational force on a moon due to a planet using Newton's law of universal gravitation. This problem involves key physics concepts such as vector operations, unit vectors, and the gravitational force formula. Ideal for high school or college-level physics students.

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