Consider the diameter of a spherical object being measured with the help of a Vernier callipers. Suppose its 10 Vernier Scale Divisions (V.S.D.) are equal to its 9 Main Scale Divisions (M.S.D.). The least division in the M.S. is 0.1 cm and the zero of V.S. is at x = 0.1 cm when the jaws of Vernier callipers are closed. If the main scale reading for the diameter is M = 5 cm and the number of coinciding vernier division is 8, the measured diameter after zero error correction, is:

Correct Option: The correct answer is 4.98 cm.

To find the measured diameter of the spherical object after correcting for zero error, we follow these step-by-step derivations:

Step 1: Calculate the Least Count (LC) of the Vernier callipers
The least count is the smallest value that can be measured accurately using the callipers. It is defined as:
$\text{Least Count (LC)}=1\text{ MSD}-1\text{ VSD}$
where MSD is the Main Scale Division and VSD is the Vernier Scale Division.

We are given that 10 Vernier Scale Divisions are equal to 9 Main Scale Divisions:
$10\text{ VSD}=9\text{ MSD}$
$1\text{ VSD}=\frac{9}{10}\text{ MSD}=0.9\text{ MSD}$

Substituting this back into the least count formula:
$\text{LC}=1\text{ MSD}-0.9\text{ MSD}=0.1\text{ MSD}$

Given that the least division on the main scale (1 MSD) is 0.1 cm:
$\text{LC}=0.1\times 0.1\text{ cm}=0.01\text{ cm}$

Step 2: Determine the Zero Error
When the jaws of the Vernier callipers are closed, the zero of the Vernier scale is at $x=0.1\text{ cm}$ (to the right of the main scale zero). This indicates a positive zero error:
$\text{Zero Error (e)}=+0.1\text{ cm}$

Step 3: Calculate the Observed Reading
The uncorrected measurement (observed reading) is given by:
$\text{Observed Reading}=\text{Main Scale Reading (M)}+\left(\text{Coinciding Vernier Division}\times \text{LC}\right)$

Given that the main scale reading $M=5\text{ cm}$ and the coinciding vernier division is 8:
$\text{Observed Reading}=5\text{ cm}+\left(8\times 0.01\text{ cm}\right)$
$\text{Observed Reading}=5\text{ cm}+0.08\text{ cm}=5.08\text{ cm}$

Step 4: Apply Zero Error Correction
The corrected reading is obtained by subtracting the zero error from the observed reading:
$\text{Corrected Reading}=\text{Observed Reading}-\text{Zero Error}$
$\text{Corrected Reading}=5.08\text{ cm}-0.1\text{ cm}=4.98\text{ cm}$

Thus, the measured diameter after zero error correction is 4.98 cm.