An object 5 cm high is placed at a distance 60 cm in fornt of a concave mirror of focal length 10 cm. Find (i) the position and (ii) size, of the image.

Answer

To solve the problem step by step, we will use the mirror formula and the magnification formula.

### Step 1: Identify the given values
- Height of the object (h₀) = 5 cm
- Object distance (u) = -60 cm (the negative sign indicates that the object is in front of the mirror)
- Focal length (f) = -10 cm (the negative sign indicates that it is a concave mirror)

### Step 2: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values into the formula:
\[ \frac{1}{-10} = \frac{1}{v} + \frac{1}{-60} \]

### Step 3: Rearranging the equation
Rearranging the equation to solve for \(\frac{1}{v}\):
\[ \frac{1}{v} = \frac{1}{-10} + \frac{1}{60} \]

### Step 4: Finding a common denominator
To add the fractions, we need a common denominator. The least common multiple of 10 and 60 is 60:
\[ \frac{1}{-10} = \frac{-6}{60} \]
So, we can rewrite the equation as:
\[ \frac{1}{v} = \frac{-6}{60} + \frac{1}{60} = \frac{-6 + 1}{60} = \frac{-5}{60} \]

### Step 5: Solving for v
Now, taking the reciprocal to find \(v\):
\[ v = \frac{60}{-5} = -12 \text{ cm} \]
This means the image is located 12 cm in front of the mirror.

### Step 6: Calculate the magnification
The magnification (M) is given by the formula:
\[ M = \frac{h_i}{h_0} = -\frac{v}{u} \]
Substituting the values:
\[ M = -\frac{-12}{-60} = \frac{12}{60} = 0.2 \]

### Step 7: Finding the height of the image
Now, we can find the height of the image (hᵢ):
\[ h_i = M \times h_0 = 0.2 \times 5 \text{ cm} = 1 \text{ cm} \]
Since the magnification is positive, the image is upright. However, since we used the negative sign for the object distance, the image is inverted.

### Final Results
(i) The position of the image is 12 cm in front of the mirror.
(ii) The size of the image is 1 cm (inverted).

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