A physical balance has its arms of unequal length. A body weight 18 kg if kept in one pan and weight 8 kg. If kept in the other pan. The true weight of the body is

Options

A. 13 kg
B. 12 kg
C. 10 kg
D. 16 kg

Answer

To find the true weight of a body using a physical balance with unequal arms, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a physical balance with unequal arms. In one pan, we have a weight of 18 kg, and in the other pan, we have a weight of 8 kg. We need to find the true weight of the body.

2. **Convert Mass to Weight**:
- The weight (W) is given by the formula \( W = m \cdot g \), where \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).
- For the 18 kg mass:
\[ W_1 = 18 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 180 \, \text{N} \]
- For the 8 kg mass:
\[ W_2 = 8 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 80 \, \text{N} \]

3. **Set Up the Torque Equation**:
- Let \( l_1 \) be the length of the arm on the side with the 18 kg weight and \( l_2 \) be the length of the arm on the side with the 8 kg weight.
- The torques around the pivot (fulcrum) must balance:
\[ W_1 \cdot l_1 = W_2 \cdot l_2 \]
- Substituting the weights:
\[ 180 \cdot l_1 = W \cdot l_2 \]
- Where \( W \) is the true weight of the body we are trying to find.

4. **Use the Second Condition**:
- When the 8 kg weight is on the other pan, we can write:
\[ W \cdot l_1 = 80 \cdot l_2 \]

5. **Set Up the Ratios**:
- From the first equation:
\[ \frac{W}{180} = \frac{l_1}{l_2} \quad \text{(1)} \]
- From the second equation:
\[ \frac{W}{80} = \frac{l_2}{l_1} \quad \text{(2)} \]

6. **Combine the Equations**:
- From (1) and (2), we can set them equal to each other:
\[ \frac{W}{180} = \frac{80}{W} \]
- Cross-multiplying gives:
\[ W^2 = 180 \cdot 80 \]
- Calculating the right side:
\[ W^2 = 14400 \]
- Taking the square root:
\[ W = \sqrt{14400} = 120 \, \text{N} \]

7. **Convert Weight Back to Mass**:
- To find the mass corresponding to this weight:
\[ m = \frac{W}{g} = \frac{120 \, \text{N}}{10 \, \text{m/s}^2} = 12 \, \text{kg} \]

### Final Answer:
The true weight of the body is **12 kg**.

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