The gravitational force between two objects of masses 3m and 5m separated by a distance 2d is 2 units. If two objects of masses 5m and 7m is separated by a distance 3d then force between them will be.

Options

A. Between 1 and 2 units
B. Below 1 unit
C. Above 2 units
D. Equal to 2 units

Answer

To solve the problem step by step, we will use the formula for gravitational force, which is given by:

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

where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the distance between the centers of the two masses.

### Step 1: Calculate the gravitational force between the first two objects

We know that the gravitational force between masses \( 3m \) and \( 5m \) separated by a distance \( 2d \) is given as 2 units.

Using the gravitational force formula:

\[ F = \frac{G \cdot (3m) \cdot (5m)}{(2d)^2} \]

This simplifies to:

\[ F = \frac{15Gm^2}{4d^2} \]

We know that this force is equal to 2 units:

\[ \frac{15Gm^2}{4d^2} = 2 \]

### Step 2: Solve for \( \frac{Gm^2}{d^2} \)

To find \( \frac{Gm^2}{d^2} \), we can rearrange the equation:

\[ 15Gm^2 = 8d^2 \]

Thus,

\[ \frac{Gm^2}{d^2} = \frac{8}{15} \]

### Step 3: Calculate the gravitational force between the second pair of objects

Now we need to find the gravitational force between the masses \( 5m \) and \( 7m \) separated by a distance \( 3d \).

Using the gravitational force formula again:

\[ F' = \frac{G \cdot (5m) \cdot (7m)}{(3d)^2} \]

This simplifies to:

\[ F' = \frac{35Gm^2}{9d^2} \]

### Step 4: Substitute \( \frac{Gm^2}{d^2} \)

Now we can substitute \( \frac{Gm^2}{d^2} = \frac{8}{15} \) into the equation for \( F' \):

\[ F' = \frac{35 \cdot \frac{8}{15}}{9} \]

### Step 5: Simplify the expression

Calculating this gives:

\[ F' = \frac{280}{135} \]

This can be simplified:

\[ F' = \frac{56}{27} \]

### Step 6: Approximate the value

Calculating \( \frac{56}{27} \) gives approximately:

\[ F' \approx 2.07 \text{ units} \]

### Final Answer

The gravitational force between the two objects of masses \( 5m \) and \( 7m \) separated by a distance \( 3d \) is approximately **2.07 units**.

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