To solve the problem regarding the vernier calipers, we need to analyze the information provided step by step. The key points to note are the positions of the scales when the jaws are closed and how they align with each other. Let's break it down systematically.
Understanding the Setup
In a vernier caliper, the main scale and the vernier scale are used to measure lengths with high precision. The main scale is marked in millimeters, while the vernier scale allows for readings between those millimeter marks.
Given Information
- The zero of the main scale coincides with the zero of the vernier scale when the jaws are closed.
- The zero of the vernier scale is slightly to the left of the 4th main scale division.
- The 5th division of the vernier scale matches with the x-th division of the main scale.
- The 9th division of the main scale matches with the 10th division of the vernier scale.
Analyzing the Measurements
From the information, we can infer the following:
- Since the zero of the vernier scale is slightly left of the 4th division of the main scale, this indicates a negative zero error. This means that the vernier scale reads slightly less than the actual measurement when the jaws are closed.
- The 5th division of the vernier scale coinciding with the x-th division of the main scale gives us a direct relationship between the two scales.
- The 9th division of the main scale coinciding with the 10th division of the vernier scale helps us determine the least count of the vernier scale.
Calculating the Least Count
The least count (LC) of the vernier caliper can be calculated using the formula:
LC = Value of 1 main scale division - Value of 1 vernier scale division
Assuming the value of 1 main scale division (MSD) is 1 mm, and since the 9th division of the main scale matches with the 10th division of the vernier scale, we can set up the equation:
9 mm = 10 × Value of 1 vernier scale division (VSD)
From this, we can find:
Value of 1 vernier scale division = 9 mm / 10 = 0.9 mm
Now, the least count can be calculated as:
LC = 1 mm - 0.9 mm = 0.1 mm
Finding the Value of x
Now that we have the least count, we can use the information about the 5th division of the vernier scale matching with the x-th division of the main scale. Since each division on the vernier scale represents 0.1 mm, the reading at the 5th division of the vernier scale is:
5 × 0.1 mm = 0.5 mm
This means that the x-th division of the main scale must be:
x mm = 0.5 mm
Thus, we can conclude that:
x = 5
Final Thoughts
In summary, the value of x, which represents the division of the main scale that corresponds to the 5th division of the vernier scale, is 5. This problem illustrates the importance of understanding how the scales interact and how to calculate measurements accurately using vernier calipers.
