If masses of two point objects are tripled and distance between them is doubled , then gravitational force of attraction between them will

Options

A. Increases by 225 %
B. Decrease by 56%
C. Increases by 125%
D. Decreases by 1445

Answer

To solve the problem step by step, we will use the universal law of gravitation, which states that the gravitational force \( F \) between two point masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by the formula:

\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

where \( G \) is the gravitational constant.

### Step 1: Identify the initial conditions
Let the initial masses of the two objects be \( m_1 \) and \( m_2 \), and the initial distance between them be \( r \).

### Step 2: Write the initial gravitational force
Using the formula for gravitational force, the initial force \( F \) is:

\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

### Step 3: Determine the new conditions
According to the problem, the masses are tripled, and the distance is doubled. Therefore, the new masses will be:

\[ m_1' = 3m_1 \quad \text{and} \quad m_2' = 3m_2 \]

The new distance will be:

\[ r' = 2r \]

### Step 4: Write the new gravitational force
Now, using the new values, the new gravitational force \( F' \) can be calculated as follows:

\[ F' = \frac{G \cdot (3m_1) \cdot (3m_2)}{(2r)^2} \]

### Step 5: Simplify the new gravitational force
Substituting the values into the equation:

\[ F' = \frac{G \cdot 9m_1 \cdot m_2}{4r^2} \]

### Step 6: Relate the new force to the initial force
We can express \( F' \) in terms of the initial force \( F \):

\[ F' = \frac{9}{4} \cdot \frac{G \cdot m_1 \cdot m_2}{r^2} = \frac{9}{4} F \]

### Step 7: Determine the change in gravitational force
To find the change in gravitational force, we can calculate the ratio of the new force to the initial force:

\[ \frac{F'}{F} = \frac{9}{4} \]

This indicates that the new force is \( 2.25 \) times the initial force.

### Step 8: Calculate the percentage increase
To find the percentage increase in gravitational force:

\[ \text{Percentage Increase} = \left( \frac{F' - F}{F} \right) \times 100 = \left( \frac{\frac{9}{4}F - F}{F} \right) \times 100 \]

\[ = \left( \frac{\frac{9}{4}F - \frac{4}{4}F}{F} \right) \times 100 = \left( \frac{5}{4}F}{F} \right) \times 100 = 125\% \]

### Final Answer
Thus, the gravitational force of attraction between the two objects will increase by **125%**.

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