The diameter of a cylinder is measured using a vernier calipers with no zero error. It is found that the zero of the vernier scale lies between 5.10 cm and 5.15 cm of the main scale. the vernier scale has 50 divisions equivalent to 2.45 cm The 24th division of the vernier scale exactly coincides with one of the main scale divisions. Calculate the diameter of a cylinder.
| Property | Description |
| Measurement Tool | Vernier Caliper |
| Object | Cylinder |
| Caliper Property | * Main scale with known divisions (MSD) * Vernier scale with known number of divisions (VSD) and total length |
| Zero Error | No zero error (vernier scale’s zero aligns with the main scale’s zero) |
| Information Provided | * Number of divisions on the main scale corresponding to the cylinder’s diameter (between two main scale divisions) * Number of vernier scale divisions that coincide with a main scale division |
| Solved Property | Diameter of the cylinder |
The task is to measure the diameter of a cylindrical object using Vernier calipers, which provides high-precision measurements. The Vernier calipers consist of two scales: the main scale and the Vernier scale. The main scale typically has a fixed gradation, and the Vernier scale allows for readings of fractions of the main scale divisions, enhancing the precision of the measurement.
Another problem on vernier calipers
Explanation:
To determine the diameter of the cylinder with Vernier calipers, it’s crucial to understand the concept of least count (LC), which is the smallest length that can be measured with the caliper and is the key to the instrument’s precision. The least count is calculated using the information about the main scale divisions (MSD) and Vernier scale divisions (VSD). In this case, the calipers’ 50 Vernier divisions are equivalent to 2.45 cm on the main scale, and the 24th Vernier division aligns precisely with a main scale division when measuring the diameter of the cylinder.
Calculation Steps:
- Calculate One Main Scale Division (MSD): The difference between two main scale readings provided (5.15 cm and 5.10 cm) is taken to find the value of one MSD, which is 0.05 cm.
- Calculate One Vernier Scale Division (VSD): The length covered by 50 Vernier scale divisions is divided by the number of divisions to find the value of one VSD, which is \( \dfrac{2.45 cm}{50} = 0.049 cm \).
- Determine the Least Count (LC): The LC of the Vernier calipers is the difference between the value of one MSD and one VSD, yielding \( 0.05 cm – 0.049 cm = 0.001 cm \).
- Calculate the Vernier Scale Reading: Multiply the LC by the number of the coinciding Vernier scale division to get the Vernier scale reading. In this case, it’s \( 24 \times 0.001 cm \).
- Determine the Diameter of the Cylinder: Add the Vernier scale reading to the main scale reading (5.10 cm) to obtain the total measurement for the diameter.
Final Answer:
The diameter of the cylinder, as measured by the Vernier calipers, is the sum of the initial main scale reading and the product of the coinciding Vernier division and the LC, which is ( 5.10 cm + 24 \times 0.001 cm = 5.124 cm ). This calculation yields the precise diameter of the cylinder.
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