Two bodies A, B of masses `m_(1), m_(2)` are knotted to a mass less string at different points rotated along concentric circles in horizontal plane. The distances of A.B from common centre are 50cm, 1m. If the tensions in the string between centre to A and A to B are in the ratio 5:4, then the ratio of `m_(1)` to `m_(2)` is

Options

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

Answer

To solve the problem, we need to analyze the forces acting on the two masses A and B, and use the given information about the tensions in the string. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the Variables
Let:
- Mass of A = \( m_1 \)
- Mass of B = \( m_2 \)
- Distance of A from the center = \( r_1 = 0.5 \) m (50 cm)
- Distance of B from the center = \( r_2 = 1 \) m
- Tension in the string from the center to A = \( T_1 \)
- Tension in the string from A to B = \( T_2 \)

### Step 2: Write the Equations for Tensions
From the problem, we know the ratio of tensions:
\[ \frac{T_1}{T_2} = \frac{5}{4} \]
This implies:
\[ T_1 = \frac{5}{4} T_2 \]

### Step 3: Apply Centripetal Force Equations
For mass A (at distance \( r_1 \)):
The net centripetal force acting on mass A is provided by the tension \( T_1 \) minus the tension \( T_2 \):
\[ T_1 - T_2 = m_1 \omega^2 r_1 \]
Substituting \( T_1 \):
\[ \frac{5}{4} T_2 - T_2 = m_1 \omega^2 (0.5) \]
This simplifies to:
\[ \frac{1}{4} T_2 = m_1 \omega^2 (0.5) \]

For mass B (at distance \( r_2 \)):
The centripetal force acting on mass B is provided solely by the tension \( T_2 \):
\[ T_2 = m_2 \omega^2 r_2 \]
Substituting \( r_2 \):
\[ T_2 = m_2 \omega^2 (1) \]

### Step 4: Substitute \( T_2 \) in the Equation
Now we can substitute \( T_2 \) from the second equation into the first equation:
\[ \frac{1}{4} (m_2 \omega^2) = m_1 \omega^2 (0.5) \]
Cancelling \( \omega^2 \) from both sides (assuming \( \omega \neq 0 \)):
\[ \frac{1}{4} m_2 = 0.5 m_1 \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ m_1 = \frac{1}{4} m_2 \cdot \frac{1}{0.5} = \frac{1}{4} m_2 \cdot 2 = \frac{1}{2} m_2 \]

### Step 6: Find the Ratio of Masses
Thus, the ratio of \( m_1 \) to \( m_2 \) is:
\[ \frac{m_1}{m_2} = \frac{1}{2} \]
This can be expressed as:
\[ m_1 : m_2 = 1 : 2 \]

### Conclusion
The ratio of the masses \( m_1 \) to \( m_2 \) is \( 1 : 2 \).

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