To solve the problem, we will follow these steps:
### Step 1: Understand the setup
We have a spherical mirror placed 10 cm below the water surface. A point object is placed 30 cm above the water surface. The refractive index of water is given as \( \mu = \frac{4}{3} \).
### Step 2: Determine the effective object distance for the mirror
The height of the object above the water surface is 30 cm. However, due to the refraction at the water surface, the effective height of the object as seen by the mirror (denoted as \( h' \)) will be modified by the refractive index of water.
Using the formula for the shift in height:
\[
h' = \mu \cdot h
\]
where \( h = 30 \, \text{cm} \) and \( \mu = \frac{4}{3} \):
\[
h' = \frac{4}{3} \cdot 30 = 40 \, \text{cm}
\]
### Step 3: Calculate the total object distance from the mirror
The distance from the water surface to the mirror is 10 cm. Therefore, the total distance \( U \) from the mirror to the object is:
\[
U = h' + \text{distance from water to mirror} = 40 \, \text{cm} + 10 \, \text{cm} = 50 \, \text{cm}
\]
Since the object is in front of the mirror, we take \( U \) as negative:
\[
U = -50 \, \text{cm}
\]
### Step 4: Determine the image distance
The images formed by the partial reflection at the water surface and the reflection from the mirror coincide. The image formed by the water surface is at a distance of 40 cm below the water surface. Since the mirror is 10 cm below the water surface, the distance from the mirror to the image \( V \) is:
\[
V = 40 \, \text{cm} - 10 \, \text{cm} = 30 \, \text{cm}
\]
Since the image is formed on the same side as the object, we take \( V \) as positive:
\[
V = 30 \, \text{cm}
\]
### Step 5: Apply the mirror formula
The mirror formula is given by:
\[
\frac{1}{f} = \frac{1}{V} + \frac{1}{U}
\]
Substituting the values of \( V \) and \( U \):
\[
\frac{1}{f} = \frac{1}{30} - \frac{1}{50}
\]
### Step 6: Calculate the focal length
To calculate \( \frac{1}{f} \):
\[
\frac{1}{f} = \frac{5 - 3}{150} = \frac{2}{150} = \frac{1}{75}
\]
Thus, the focal length \( f \) is:
\[
f = 75 \, \text{cm}
\]
### Final Answer
The focal length of the mirror is \( 75 \, \text{cm} \).
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