Two point masses m and M are separated by a distance d. If the separation d remains fixed and the masses are increased to the values 3m and 3M, respectively, how does the gravitational force between them change ?

Options

A. The force will be one-third as great.
B. The force will be one-ninth as great.
C. The force will be three times as great.
D. The force will be nine times as great.

Answer

To solve the problem, we will use Newton's law of universal gravitation, which states that the gravitational force \( F \) between two point masses \( m \) and \( M \) separated by a distance \( d \) is given by the formula:

\[ F = G \frac{mM}{d^2} \]

where \( G \) is the gravitational constant.

### Step 1: Write the initial gravitational force
Initially, the gravitational force \( F_1 \) between the masses \( m \) and \( M \) is:

\[ F_1 = G \frac{mM}{d^2} \]

### Step 2: Determine the new masses
The problem states that the masses are increased to \( 3m \) and \( 3M \).

### Step 3: Write the new gravitational force
The new gravitational force \( F_2 \) between the masses \( 3m \) and \( 3M \) is:

\[ F_2 = G \frac{(3m)(3M)}{d^2} \]

### Step 4: Simplify the new gravitational force
Now, simplify \( F_2 \):

\[ F_2 = G \frac{9mM}{d^2} \]

### Step 5: Compare the new force with the initial force
Now, we can compare \( F_2 \) with \( F_1 \):

\[ F_2 = 9 \left( G \frac{mM}{d^2} \right) = 9 F_1 \]

### Conclusion
Thus, the gravitational force between the two masses increases by a factor of 9 when both masses are increased to three times their original values while keeping the distance \( d \) constant. Therefore, the gravitational force becomes 9 times greater.

### Final Answer
The gravitational force between the two masses becomes **9 times greater**.

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