Question

Plot the graph between \( \frac{1}{u} \) and \( \frac{1}{v} \) for the given concave mirror and then evaluate focal length by graph.

Hint:

When drawing the graph, choose scales that utilize at least two-thirds of the graph paper area for better accuracy. The best-fit line should be drawn such that there is an equal distribution of points on either side of the line. Do not force the line to pass through the origin.

Answer 1

Step 1: Understanding the Concept: 
This experiment is a graphical method to determine the focal length of a concave mirror. It uses the data of object distance (u) and image distance (v) obtained from the two-pin method. By plotting \( \frac{1}{v} \) against \( \frac{1}{u} \), we can use the intercepts of the resulting straight-line graph to calculate the focal length, which often provides a more accurate result than averaging individual calculations. 
Step 2: Key Formula and Apparatus: 
Apparatus Required: 
The same as for the two-pin method: an optical bench, a concave mirror, a mirror holder, two optical needles (pins), and a meter scale. Also, a graph paper is needed. 
Key Formula: 
The mirror formula is: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] This can be rearranged into the form of a straight-line equation \( y = mx + c \): \[ \frac{1}{v} = -\frac{1}{u} + \frac{1}{f} \] Comparing this with \( y = mx + c \), we have: - \( y = \frac{1}{v} \) - \( x = \frac{1}{u} \) - Slope \( m = -1 \) - Y-intercept \( c = \frac{1}{f} \) 
When \( \frac{1}{v} = 0 \), we get the X-intercept as \( \frac{1}{u} = \frac{1}{f} \). 
Step 3: Detailed Procedure: 
1. Data Collection: 
- Perform the experiment to find the focal length of the concave mirror using the two-pin method (as in Question 1). 
- Obtain at least 5-6 sets of readings for object distance (u) and the corresponding image distance (v). 
2. Data Processing: 
- For each pair of (u, v), calculate their reciprocals: \( x = \frac{1}{u} \) and \( y = \frac{1}{v} \). 
- Tabulate the results. 
 

3. Plotting the Graph: 
- Choose a suitable scale for both axes on the graph paper. 
- Plot the graph with \( \frac{1}{u} \) along the X-axis and \( \frac{1}{v} \) along the Y-axis. 
- The plotted points should lie on a straight line. Draw the best-fit straight line passing through these points. 
- The line will have a negative slope and will intersect both the positive X and Y axes. 

Step 4: Calculation from Graph and Final Answer: 
1. Finding Intercepts: 
- Find the Y-intercept (OA) where the line cuts the Y-axis (\( \frac{1}{u} = 0 \)). 
- Find the X-intercept (OB) where the line cuts the X-axis (\( \frac{1}{v} = 0 \)). 
2. Calculating Focal Length: 
- From the Y-intercept: \( OA = \frac{1}{f} \implies f_1 = \frac{1}{OA} \). 
- From the X-intercept: \( OB = \frac{1}{f} \implies f_2 = \frac{1}{OB} \). 
- The focal length of the mirror is the mean of these two values: \[ f = \frac{f_1 + f_2}{2} \] The result is stated as: "The focal length of the concave mirror as determined from the graph is f cm."