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Question: The main scale of a vernier calipers reads in millimeter and its vernier is divided into 10 division...

The main scale of a vernier calipers reads in millimeter and its vernier is divided into 10 division which coincide with 9 division of main scale. When two jaws of instrument touch each other the fourth division of the Vernier scale coincides with a main scale division and the zero of the vernier lies to the right of zero of main scale and lies Between 2nd and 3rd division of main scale. Further more, when a cylinder is tightly placed along its length between two jaws, it is observed that 78th division of the main scale coincides sixth division of vernier scale, calculate the length of the cylinder in cm.

A

3.14 cm

B

3.07 cm

C

3.2 cm

D

3.04 cm

Answer

7.62 cm

Explanation

Solution

Solution:

  1. Least Count (LC):
    Since 10 vernier divisions = 9 main scale divisions, one main scale division = 1 mm and one vernier division
    = (9 mm / 10) = 0.9 mm.
    Thus,

    LC=1 mm−0.9 mm=0.1 mm=0.01 cm.\text{LC} = 1\,\text{mm} - 0.9\,\text{mm} = 0.1\,\text{mm} = 0.01\,\text{cm}.

  2. Zero Error:
    When the jaws are closed, the vernier zero does not coincide with main zero. Instead, the 4th vernier division lines up with a main scale division and the “0” of the vernier lies between the 2nd and 3rd main scale division.
    This means the main scale reading just before the vernier “0” is 2 × 0.1 cm = 0.2 cm, and the extra reading due to the 4th division is

    4×LC=4×0.01 cm=0.04 cm.4 \times \text{LC} = 4 \times 0.01\,\text{cm} = 0.04\,\text{cm}.

    Thus, the zero error = 0.2 cm + 0.04 cm = 0.24 cm (a positive error).

  3. Measured Length of Cylinder:
    When measuring the cylinder, the main scale shows the 78th division and the 6th vernier division coincides. So,

    Main scale reading=78 mm=7.8 cm,\text{Main scale reading} = 78\,\text{mm} = 7.8\,\text{cm}, Vernier reading=6×LC=6×0.01 cm=0.06 cm.\text{Vernier reading} = 6 \times \text{LC} = 6 \times 0.01\,\text{cm} = 0.06\,\text{cm}.

    Measured length = 7.8 cm + 0.06 cm = 7.86 cm.

  4. True Length:
    Subtract the zero error from the measured reading:

    True length=7.86 cm−0.24 cm=7.62 cm.\text{True length} = 7.86\,\text{cm} - 0.24\,\text{cm} = 7.62\,\text{cm}.

Explanation (Core Steps):

  • LC = 0.01 cm.
  • Zero error = (0.2 + 0.04) cm = 0.24 cm.
  • Measured reading with cylinder = 7.8 + 0.06 = 7.86 cm.
  • True reading = 7.86 – 0.24 = 7.62 cm.