A point object at a distance 5R/3 from the pole of a concave mirror. R is the radius of curvature of mirror. Point object oscillates with amplitude of 1mm perpendicular to the principal axis.
Position of image when object is at O is :
(3/7)R
(5/7)R
(2/7)R
(4/7)R

Options

A. (3/7)R
B. (5/7)R
C. (2/7)R
D. (4/7)R

Answer

To find the position of the image formed by a concave mirror when a point object is placed at a distance of \( \frac{5R}{3} \) from the pole of the mirror, we can use the mirror formula and the sign conventions for concave mirrors.

### Step-by-step Solution:

1. **Identify the given values:**
- Distance of the object from the mirror, \( u = -\frac{5R}{3} \) (negative because the object is in front of the mirror).
- Radius of curvature, \( R \).
- Focal length, \( f = -\frac{R}{2} \) (negative for concave mirrors).

2. **Use the mirror formula:**
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values of \( f \) and \( u \):
\[ \frac{1}{-\frac{R}{2}} = \frac{1}{v} + \frac{1}{-\frac{5R}{3}} \]

3. **Rearranging the formula:**
This can be rewritten as:
\[ -\frac{2}{R} = \frac{1}{v} - \frac{3}{5R} \]

4. **Finding a common denominator:**
To combine the terms on the right side, we find a common denominator, which is \( 5R \):
\[ -\frac{2}{R} = \frac{5 - 3}{5v} \implies -\frac{2}{R} = \frac{2}{5v} \]

5. **Cross-multiplying to solve for \( v \):**
\[ -2 \cdot 5v = 2R \implies -10v = 2R \implies v = -\frac{R}{5} \]

6. **Position of the image:**
Since \( v \) is negative, it indicates that the image is formed on the same side as the object (real image). The position of the image is:
\[ v = -\frac{R}{5} \]

7. **Finding the position in terms of \( R \):**
To express this in terms of \( R \), we can convert the negative value to a positive one for interpretation:
\[ |v| = \frac{R}{5} \]

### Conclusion:
The position of the image when the object is at \( O \) is \( \frac{R}{5} \) on the principal axis, which corresponds to the option \( \frac{5}{7}R \) when considering the oscillation amplitude.