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 find the correct weight of the object using a beam balance of unequal arm lengths, we can follow these steps:

### Step 1: Understand the System
We have a beam balance with two pans of unequal lengths. Let:
- \( W \) = actual weight of the object
- \( W_1 \) = weight measured when the object is on the left pan
- \( W_2 \) = weight measured when the object is on the right pan
- \( r_1 \) = length of the left arm
- \( r_2 \) = length of the right arm

### Step 2: Set Up the Moment Equations
When the object is placed on the left pan, the balance is in equilibrium, which gives us:
\[ W \cdot r_1 = W_1 \cdot r_2 \quad \text{(1)} \]
This equation states that the moment produced by the actual weight on the left side is equal to the moment produced by the weight \( W_1 \) on the right side.

When the object is placed on the right pan, we have:
\[ W_2 \cdot r_1 = W \cdot r_2 \quad \text{(2)} \]
This equation states that the moment produced by the weight \( W_2 \) on the left side is equal to the moment produced by the actual weight \( W \) on the right side.

### Step 3: Solve the Equations
From equation (1):
\[ W = \frac{W_1 \cdot r_2}{r_1} \]
From equation (2):
\[ W = \frac{W_2 \cdot r_1}{r_2} \]

### Step 4: Equate the Two Expressions for W
Setting the two expressions for \( W \) equal to each other:
\[ \frac{W_1 \cdot r_2}{r_1} = \frac{W_2 \cdot r_1}{r_2} \]

### Step 5: Cross Multiply
Cross multiplying gives us:
\[ W_1 \cdot r_2^2 = W_2 \cdot r_1^2 \]

### Step 6: Express the Actual Weight
Now, we can express the actual weight \( W \) in terms of \( W_1 \) and \( W_2 \):
\[ W = \sqrt{W_1 \cdot W_2} \]

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

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