When a lens is inserted between an object and a screen which are fixed distance apart the size of the image is either `6` cm or `2/3 cm.` Find size of the object

Explanation

To solve the problem of finding the size of the object when a lens is inserted between an object and a screen, we can use the displacement method. Here’s a step-by-step solution:

### Step 1: Identify the sizes of the images
We are given two sizes of the images formed by the lens:
- \( I_1 = 6 \, \text{cm} \)
- \( I_2 = \frac{2}{3} \, \text{cm} \)

### Step 2: Use the Displacement Method Formula
According to the displacement method, the size of the object \( O \) can be calculated using the formula:
\[ O = \sqrt{I_1 \times I_2} \]

### Step 3: Substitute the values into the formula
Now, we will substitute the values of \( I_1 \) and \( I_2 \) into the formula:
\[ O = \sqrt{6 \times \frac{2}{3}} \]

### Step 4: Simplify the expression inside the square root
Calculating the multiplication:
\[ 6 \times \frac{2}{3} = \frac{12}{3} = 4 \]

### Step 5: Calculate the square root
Now, we take the square root of \( 4 \):
\[ O = \sqrt{4} = 2 \, \text{cm} \]

### Conclusion
Thus, the size of the object is:
\[ \boxed{2 \, \text{cm}} \]

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