Example 2 Clw suppase two objects M1 and m2 are a distance ol apart. Determine how the magnitude of the porce batween. them Charges if: 2.1 Both masses double and the distance between them is halved 2.2 The mass of mi doubles

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To solve the problem, we need to apply Newton's law of universal gravitation, which states that the force $$ F $$ between two masses $$ m_1 $$ and $$ m_2 $$ separated by a distance $$ r $$ is given by the formula:

$$
F = G \frac{m_1 m_2}{r^2}
$$

where $$ G $$ is the gravitational constant.

### Known Conditions:
- Initial masses: $$ m_1 $$ and $$ m_2 $$
- Initial distance: $$ r $$

### 2.1 Both masses double and the distance between them is halved

1. **New masses**: 
   - $$ m_1' = 2m_1 $$
   - $$ m_2' = 2m_2 $$

2. **New distance**: 
   - $$ r' = \frac{r}{2} $$

3. **New force**: 
   We can substitute the new values into the gravitational force formula:

$$
F' = G \frac{m_1' m_2'}{(r')^2}
$$

Substituting the new values:

$$
F' = G \frac{(2m_1)(2m_2)}{\left(\frac{r}{2}\right)^2}
$$

4. **Simplifying the expression**:

$$
F' = G \frac{4m_1 m_2}{\frac{r^2}{4}} = G \frac{4m_1 m_2 \cdot 4}{r^2} = 16 G \frac{m_1 m_2}{r^2}
$$

Thus, the new force $$ F' $$ is:

$$
F' = 16F
$$

### 2.2 The mass of $$ m_1 $$ doubles

1. **New mass**: 
   - $$ m_1' = 2m_1 $$
   - $$ m_2' = m_2 $$ (remains the same)

2. **Distance remains the same**: 
   - $$ r' = r $$

3. **New force**: 
   Substitute the new mass into the gravitational force formula:

$$
F' = G \frac{m_1' m_2'}{(r')^2}
$$

Substituting the new values:

$$
F' = G \frac{(2m_1)(m_2)}{r^2}
$$

4. **Simplifying the expression**:

$$
F' = 2G \frac{m_1 m_2}{r^2}
$$

Thus, the new force $$ F' $$ is:

$$
F' = 2F
$$

### Summary of Results:
- **2.1**: The new force when both masses double and the distance is halved is $$ 16F $$.
- **2.2**: The new force when only $$ m_1 $$ doubles is $$ 2F $$.
- **2.1**: When both masses double and the distance is halved, the force becomes 16 times the original force. - **2.2**: When only $$ m_1 $$ doubles, the force becomes 2 times the original force.

Example 2 Clw suppase two objects M1 and m2 are a distance ol apart. Determine how the magnitude of the porce batween. them Charges if: 2.1 Both masses double and the distance between them is halved 2.2 The mass of mi doubles

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