Determining the Radius of Curvature of a Given Spherical Surface by Spherometer, Steps, Sample Questions

Radius of curvature is the reciprocal of a curve at a mentioned point. A spherometer is used to determine the radius of curvature. The aim of a spherometer is to measure the curvature of a spherical surface so it can be convex or concave by using a spherometer.

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Radius of Curvature

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In mathematics, the formula for Radius of curvature is given as-

\(R=\frac{l^2}{6h} + \frac{h}{2}\)\(\)

Where h= height up to which the spherical surface has bent.

l= leg length of the spherometer (Leg length is the average length of the leg calculated using a scale.) 

r= radius of the curvature

Also Read: Surface Energy

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Few Common Terms to Understand

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Before getting into the depths of the experiment few terms are enlisted herein:-

1. Least count- The least count means the distance measured by the spherometer. The Least count Formula is:

\(LC = \frac{Pitch}{No. of Division on the circular scale}\)

2. Pitch: Pitch is the distance covered between two points of the screw. It means by giving one rotation, only one thread is advanced in the screw. So, if only one thread is advanced, then the distance covered in the main scale is the pitch. The division covered in the circular scale is 100 (0 to 90).

Now, Let's take an example to understand the process to calculate Radius of Curvature.

Assume h₁ as 1, h₂ as 2, and h₃ as 3, and the length of the three sides of the triangular surface is 3.2, 3.2, and 3.1. Calculate the Radius of the spherical surface using a tabular format.

It can be solved in following steps: 

1. By Calculating I_mean which is given as: \(I(Mean)= \frac{AB+BC+CA}{3}\)
2. No once I_mean is known, now we have to calculate the least count

The measurement should be taken using a spherometer.

1. Sagitta Table (h)

Hence,

1. If the value of h was not given in the question, then find out every value of h.
2. Mean of h= [h₁+h₂+h₃/3]= [1+2+3/3]= 2mm
3. Mean of l= [(3.2+3.2+3.1)/3] = 3.16cm

Therefore, R= {(3.1)²/6*2}+{2/2}= 1.80 cm.

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What is a Spherometer?

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It is a simple three-legged scientific device that determines the radius of curvature. The spherometer has a big middle leg. The middle big leg has a rotating screw that is circulated to coincide the edge with zero.

Spherometer

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Experiment to Determine the Procedure of Calculating Radius of Curvature Using Spherometer

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The radius of the curvature can be determined simply through a small experiment. The apparatus required is the spherometer and the watch glass. The glass is required because it has a convex or a concave spherical surface. 

Steps Involved

1. Calculate the least count of the spherometer. 
2. Use the circular and vertical scale together in a spherometer. 
3. Give 5 rotations to the circular scale and mark the reading from the circular scale. Edge is the reference point. (Give rotation in such a manner that 53 points are completed)
4. Circulate the screw and give 5 rotations.
5. You can give clockwise or anti-clockwise rotation as the only thing that gets calculated is the distance that is covered in the main scale.

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Observation of the Experiment Presented in Tabular Format

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Using the above data the below-mentioned table is presented to determine the observations.

The process of taking the observations is as follows: 

1. Take a watch glass and the procedure of convex and concave spherical surfaces are different. 
2. For such a procedure, one must consider at least 10 observations to minimize the error. Herein, only one observation is considered to explain the procedure. Similarly, 9 more observations to consider to complete the process. 
3. Observation in a convex spherical surface:
   1. Keep the watch-glass on the convex surface side and then take the spherometer and keep it above the watch-glass.
   2. Give rotation to the circular scale till the middle leg of the spherometer touches the top surface of the convex spherical surface of the watch-glass. 
   3. Further, to understand whether the same is touching the glass, take a small piece of paper to check the same. 
   4. If all the three legs start rotating, it means the middle leg has touched the above surface. 
   5. If the small paper passes from between the surface of the glass and the middle leg of the spherometer, it means the middle leg has not touched the glass yet. So, place it properly and give back rotations. 
   6. Give the exact position where the middle leg touches the surface. 
   7. Then, take the reading from the circular scale. And the edge of the main scale is the reference. 
   8. Now, suppose the main scale is coinciding with 21 points of the circular scale, then 21 is the initial circular scale reading. 
   9. Further, observe NCR. Take the spherometer and keep it on a flat surface. 
   10. A small gap lies between the middle leg of the spherometer and the flat surface. You can use a small piece of paper to check the same. 
   11. Now, rotate the circular scale till the middle leg touches the upper surface of the flat area. 
   12. Previously, it was noted that the reference point was 21. So, taking that as the base, rotate the spherometer till the middle leg touches the surface. 
   13. After the number of rotations gets completed and the middle leg touches the surface, the exact point is determined. (where the paper does not pass from between the middle leg and the surface of the flat).
   14. Then, note the reading from the circular scale edge. Suppose the reading is 4.
   15. Now, INF calculation

 if I > F, then INF is (I - F)

 Where I = Initial Circular scale reading

F= Final Circular Scale Reading

And if I < F, then add the number of divisions in the circular scale i.e. 100 and I, and deduct F 

i.e. [(100+I)-F]

1. PSR= Pitch* Number of Complete rotation
2. CSR= INF * List Counts

Therefore, in the above table,

• ICSR (assumed reading)= 21
• NCR (assumed reading)= 1
• FCSR (assumed reading)= 4
• INF equals= (21-4)= 17
• PSR= (0.1cm*1)= 0.1 cm

Where, pitch (calculated above) is 0.1

CSR= (17 * 0.001cm)= 0.17cm

Where, 17 is the INF 

0.001 is the least count (calculated above)

• Total= (O.1+0.17)= 0.117cm
• Mean is also calculated as at least 10 observations are taken so that less number of errors of minimum error is committed. 

In such a manner the measurement of the quantity (i.e. the height up to which the spherometer was raised.) 

4. Observation in a Concave Spherical Surface

For a concave spherical surface, the same process will be followed but in a reverse manner. 

Reverse Procedure:

1. First, take the Spherometer and place it on a plain surface. 
2. Give rotation to the circular scale till it touches the surface. 
3. Check the difference gap with a piece of paper in a similar manner followed in a convex surface and adjust it to get the edge reading.
4. Then, put the spherometer in its current position on the glass watch and observe the gap. 
5. Adjust the rotations in such a way so that the gap diminishes and the middle leg touches the glass surface. 
6. Simultaneously, note the number of complete rotations and the FCSR.

Also Read: Bernoulli’s principle

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Things to Remember

1. Always use a small piece of paper to determine the gap between the middle leg of the spherometer and the surface.
2. Always use at least 10 figures to determine the observations.
3. Rotate the screw smoothly. If any obstacle while rotating, then remove such obstruction.
4. Select glass watches of the same sizes.