A body of mass 1 kg is placed between two bodies of masses 10 kg and 90 kg and they are arranged in a straight line. If the forces of attraction due to the two masses on the body of mass 1 kg are equal, then the ratio of the distance between the masses 10 kg and 1 kg to the distance between the masses of 1 kg and 90 kg is _________

Options

A. ` 1: 1`
B. ` 2 : 3`
C. ` 1 : 3`
D. ` 1 : 9`

Explanation

To solve the problem, we need to find the ratio of the distances between the masses. Let's break it down step by step:

### Step 1: Understand the Problem
We have three masses:
- Mass \( m_1 = 10 \, \text{kg} \)
- Mass \( m_2 = 1 \, \text{kg} \)
- Mass \( m_3 = 90 \, \text{kg} \)

The mass \( m_2 \) is placed between \( m_1 \) and \( m_3 \). We need to find the ratio of the distance \( x \) (between \( m_1 \) and \( m_2 \)) to the distance \( y \) (between \( m_2 \) and \( m_3 \)) when the gravitational forces acting on \( m_2 \) due to \( m_1 \) and \( m_3 \) are equal.

### Step 2: Set Up the Gravitational Force Equations
The gravitational force \( F \) between two masses is given by the formula:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
Where:
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses,
- \( r \) is the distance between the masses.

For our scenario:
- The force \( F_1 \) acting on \( m_2 \) due to \( m_1 \) (10 kg) is:
\[ F_1 = \frac{G \cdot 10 \cdot 1}{x^2} \]
- The force \( F_2 \) acting on \( m_2 \) due to \( m_3 \) (90 kg) is:
\[ F_2 = \frac{G \cdot 90 \cdot 1}{y^2} \]

### Step 3: Set the Forces Equal
Since the forces are equal, we can set \( F_1 = F_2 \):
\[ \frac{G \cdot 10 \cdot 1}{x^2} = \frac{G \cdot 90 \cdot 1}{y^2} \]
We can cancel \( G \) and \( 1 \) from both sides:
\[ \frac{10}{x^2} = \frac{90}{y^2} \]

### Step 4: Cross Multiply
Cross multiplying gives us:
\[ 10 \cdot y^2 = 90 \cdot x^2 \]

### Step 5: Simplify the Equation
Dividing both sides by 10:
\[ y^2 = 9 \cdot x^2 \]

### Step 6: Take the Square Root
Taking the square root of both sides:
\[ y = 3x \]

### Step 7: Find the Ratio
Now, we can find the ratio of \( x \) to \( y \):
\[ \frac{x}{y} = \frac{x}{3x} = \frac{1}{3} \]

### Final Answer
The ratio of the distance between the masses \( 10 \, \text{kg} \) and \( 1 \, \text{kg} \) (i.e., \( x \)) to the distance between the masses \( 1 \, \text{kg} \) and \( 90 \, \text{kg} \) (i.e., \( y \)) is:
\[ \frac{x}{y} = \frac{1}{3} \]