A beam balance of unequal arm length is used by an unscruplus trader. When an object is weighed on left pan. The weight i found to be `W_(1)`. When the object is weighed on the right plane, the weight is found to be `W_(2)`. If `W_(1) ne W_(2)`, the correct weight of the object is

Options

A. `(W_(1) + W_(2))/(2)`
B. `sqrt(W_(1)W_(2))`
C. `sqrt(W_(1)^(2)+W_(2)^(2))`
D. `sqrt(W_(2)^(2)-W_(1)^(2))`

Answer

To solve the problem, we need to establish the relationship between the weights measured on the beam balance with unequal arms. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the Beam Balance
A beam balance consists of a beam that pivots at a point (the fulcrum). The lengths of the arms on either side of the fulcrum can be different, which affects the measurements of weight.

### Step 2: Setting Up the Equations
When an object is weighed on the left pan, the weight is denoted as \( W_1 \). When the object is weighed on the right pan, the weight is denoted as \( W_2 \).

1. **For the left pan**:
- Let the actual weight of the object be \( W \).
- The torque balance condition gives us:
\[ W \cdot L_1 = W_1 \cdot L_2 \quad (1) \]
Here, \( L_1 \) is the length of the arm on the left side and \( L_2 \) is the length of the arm on the right side.

2. **For the right pan**:
- The torque balance condition gives us:
\[ W_2 \cdot L_1 = W \cdot L_2 \quad (2) \]

### Step 3: Solving the Equations
From equation (1):
\[ W = \frac{W_1 \cdot L_2}{L_1} \]

From equation (2):
\[ W = \frac{W_2 \cdot L_1}{L_2} \]

### Step 4: Equating the Two Expressions for \( W \)
Since both expressions equal \( W \), we can set them equal to each other:
\[ \frac{W_1 \cdot L_2}{L_1} = \frac{W_2 \cdot L_1}{L_2} \]

### Step 5: Cross-Multiplying
Cross-multiplying gives us:
\[ W_1 \cdot L_2^2 = W_2 \cdot L_1^2 \]

### Step 6: Rearranging to Find \( W \)
Rearranging the above equation, we can express \( W \) in terms of \( W_1 \) and \( W_2 \):
\[ W^2 = W_1 \cdot W_2 \]
Thus,
\[ W = \sqrt{W_1 \cdot W_2} \]

### Conclusion
The correct weight of the object is:
\[ W = \sqrt{W_1 \cdot W_2} \]