Find the focal length of given concave mirror by using two-pin method.
Show Hint
To remove parallax effectively, ensure that the tips of the object pin, image pin, and the principal axis of the mirror are at the same horizontal level. Always keep your eye at a distance of at least 30 cm from the pin while checking for parallax.
Step 1: Understanding the Concept:
This experiment aims to determine the focal length of a concave mirror by measuring the object distance (u) and the corresponding image distance (v). The relationship between u, v, and the focal length (f) is given by the mirror formula. The "two-pin method" refers to using one pin as the object and another pin to locate the position of its real, inverted image by eliminating parallax. Step 2: Key Formula and Apparatus: Apparatus Required:
An optical bench, a concave mirror, a mirror holder, two optical needles (pins), and a meter scale. Key Formula:
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
According to sign convention for a concave mirror forming a real image:
Object distance (u) is negative.
Image distance (v) is negative.
Focal length (f) is negative.
So, the formula becomes:
\[ \frac{1}{-f} = \frac{1}{-v} + \frac{1}{-u} \implies \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
From this, the focal length can be calculated as:
\[ f = \frac{uv}{u+v} \]
Step 3: Detailed Procedure: 1. Preliminary Setup:
- First, find the approximate focal length (f) of the concave mirror by focusing the image of a distant object (like a tree) on a screen. The distance between the mirror and the screen gives a rough estimate of f.
- Mount the concave mirror on the holder and place it on the optical bench.
- Place one pin (the object pin, O) in front of the mirror at a distance approximately 1.5f from the mirror. 2. Locating the Image:
- Place the second pin (the image pin, I) on the other side of the object pin.
- Adjust the position of the image pin until there is no parallax between the tip of the image pin and the tip of the inverted image of the object pin. Parallax is removed when the image pin and the image of the object pin do not separate when you move your eye side-to-side. 3. Taking Measurements:
- Record the position of the mirror, the object pin (O), and the image pin (I) on the optical bench scale.
- Calculate the object distance, \( u = \text{position of mirror} - \text{position of object pin} \).
- Calculate the image distance, \( v = \text{position of mirror} - \text{position of image pin} \). 4. Repeating the Experiment:
- Repeat the experiment for at least 4-5 different values of u, by moving the object pin further away from the mirror. For each position of u, locate the corresponding image position v. 5. Observation Table: Step 4: Calculation and Final Answer:
For each set of u and v, calculate the focal length f using the formula \( f = \frac{uv}{u+v} \).
Finally, calculate the mean value of the focal length from all the readings.
\[ f_{mean} = \frac{f_1 + f_2 + f_3 + ...}{n} \]
The mean value is the focal length of the given concave mirror.
