A 1 cm object is placed perpendicular to the principal axis of a convex mirror of focal length 7.5 cm. Find its distance from the mirror if the image formed is 0.6 cm in size.

Explanation

To solve the problem step by step, we will use the concepts of magnification and the mirror formula for a convex mirror.

### Step 1: Identify the given values
- Height of the object (ho) = 1 cm
- Height of the image (hi) = 0.6 cm
- Focal length of the convex mirror (f) = 7.5 cm

### Step 2: Write the magnification formula
The magnification (m) for mirrors is given by the formula:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]
where:
- \( h_i \) = height of the image
- \( h_o \) = height of the object
- \( v \) = image distance
- \( u \) = object distance

### Step 3: Substitute the known values into the magnification formula
Substituting the values of \( h_i \) and \( h_o \):
\[ \frac{0.6}{1} = -\frac{v}{u} \]
This simplifies to:
\[ 0.6 = -\frac{v}{u} \]

### Step 4: Rearrange to find the relationship between v and u
From the above equation, we can express \( v \) in terms of \( u \):
\[ v = -0.6u \]
Let this be our Equation 1.

### Step 5: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( f = 7.5 \) cm and \( v = -0.6u \) into the mirror formula:
\[ \frac{1}{7.5} = \frac{1}{-0.6u} + \frac{1}{u} \]

### Step 6: Simplify the equation
To simplify, we need to find a common denominator for the right side:
\[ \frac{1}{7.5} = \frac{-1 + 0.6}{0.6u} \]
This simplifies to:
\[ \frac{1}{7.5} = \frac{-0.4}{0.6u} \]

### Step 7: Cross-multiply to solve for u
Cross-multiplying gives:
\[ 0.4 = \frac{0.6u}{7.5} \]
Now, multiplying both sides by 7.5:
\[ 0.4 \times 7.5 = 0.6u \]
\[ 3 = 0.6u \]

### Step 8: Solve for u
Now, divide both sides by 0.6:
\[ u = \frac{3}{0.6} = 5 \text{ cm} \]

### Step 9: Determine the sign of u
Since the object is placed in front of the mirror, the object distance \( u \) is taken as negative in the sign convention for mirrors:
\[ u = -5 \text{ cm} \]

### Final Answer
The distance of the object from the mirror is \( 5 \text{ cm} \) (considering the sign convention).

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