A marble is dropped straight down from a distance h above the floor.
Let `F_(m)` = the magnitude of the gravitational force on the marble due to the Earth,
`F_(e)` = the magnitude of the gravitational force on the Earth due to the marble,
`a_(m)` = the magnitude of the acceleration of the marble toward the Earth,
`a_(e)` = the magnitude of the acceleration of the Earth toward the marble.
Which set of conditions is true as the marble falls toward the earth? Neglect any effects of air resistance.

Options

A. `F_(m)=F_(e)` and `a_(m) lt a_(e)`
B. `F_(m) lt F_(e)` and `a_(m) gt a_(e)`
C. `F_(m) lt F_(e)` and `a_(m) = a_(e)`
D. `F_(m=F_(e)` and `a_(m) gt a_(e)`

Answer

To solve the problem, we will analyze the forces acting on the marble and the Earth as the marble falls.

### Step-by-Step Solution:

1. **Identify the Forces**:
- The gravitational force acting on the marble due to the Earth is denoted as \( F_m \).
- The gravitational force acting on the Earth due to the marble is denoted as \( F_e \).

2. **Apply Newton's Third Law**:
- According to Newton's Third Law, every action has an equal and opposite reaction. Therefore, the magnitudes of the forces are equal:
\[ F_m = F_e \]

3. **Calculate the Gravitational Force on the Marble**:
- The gravitational force acting on the marble is given by:
\[ F_m = m \cdot g \]
- Here, \( g \) is the acceleration due to gravity, which can be approximated as \( g = \frac{GM}{R^2} \) where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

4. **Calculate the Gravitational Force on the Earth**:
- The gravitational force acting on the Earth due to the marble is:
\[ F_e = M \cdot a_e \]
- Here, \( a_e \) is the acceleration of the Earth toward the marble.

5. **Set the Forces Equal**:
- Since \( F_m = F_e \), we can equate the two expressions:
\[ m \cdot g = M \cdot a_e \]

6. **Determine the Acceleration of the Earth**:
- Rearranging the equation gives:
\[ a_e = \frac{m \cdot g}{M} \]
- Since \( M \) (mass of the Earth) is much larger than \( m \) (mass of the marble), \( a_e \) will be very small compared to \( g \).

7. **Determine the Acceleration of the Marble**:
- The acceleration of the marble toward the Earth is simply \( g \):
\[ a_m = g \]

8. **Conclusion**:
- The magnitudes of the gravitational forces \( F_m \) and \( F_e \) are equal.
- The acceleration of the marble \( a_m \) is much greater than the acceleration of the Earth \( a_e \):
\[ a_m > a_e \]
- Therefore, the correct conclusion is that the magnitude of both forces is equal, but the acceleration of the Earth is much less than that of the marble.

### Final Answer:
- The magnitude of the gravitational force on the marble due to the Earth \( F_m \) is equal to the magnitude of the gravitational force on the Earth due to the marble \( F_e \).
- The acceleration of the marble \( a_m \) is much greater than the acceleration of the Earth \( a_e \).