The pitch of screw gauge is `1mm` and its circular scale is divided into 100 divisions. When nothing is put between the studs, the zero of main scale is not seen, but when circular scale is rotated by `450^(@)` the zero of main scale is seen and the zero of mai scale coincides with the zero of circular scale. When a glass plate is placed betwen the studs, the ciruclar scale lies beteen `18^(th)` and `19^(th)` division of main scale and circular scale reads 34 divisions. Then

Options

A. Error is positive zero error and its magnitude is `1.25mm`
B. Error is negative zero error and its magnitude is `1.25mm`
C. The thickness of the glass plate is `19.59mm`
D. The thickness of the glass plate is `17.09mm`

Answer

To solve the problem step by step, we will follow the outlined procedure to find the thickness of the glass plate and the zero error of the screw gauge.

### Step 1: Calculate the Least Count of the Screw Gauge
The least count (LC) of the screw gauge can be calculated using the formula:
\[ \text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}} \]
Given:
- Pitch = 1 mm
- Number of divisions = 100

Substituting the values:
\[ \text{Least Count} = \frac{1 \text{ mm}}{100} = 0.01 \text{ mm} \]

### Step 2: Determine the Negative Zero Error
When nothing is placed between the studs, the zero of the main scale is not visible, indicating a negative zero error. The circular scale is rotated by \(450^\circ\), which can be broken down as follows:
- \(450^\circ = 360^\circ + 90^\circ\)

The movement of the screw gauge head for one complete rotation (360°) is equal to the pitch (1 mm). For \(90^\circ\) (which is \(\frac{1}{4}\) of a rotation), the movement will be:
\[ \text{Movement for } 90^\circ = \frac{1 \text{ mm}}{4} = 0.25 \text{ mm} \]
Thus, the total negative zero error (Ze) is:
\[ \text{Negative Zero Error} = -\left(1 \text{ mm} + 0.25 \text{ mm}\right) = -1.25 \text{ mm} \]

### Step 3: Read the Main Scale and Circular Scale
When a glass plate is placed between the studs:
- The main scale reading (MSR) lies between the 18th and 19th divisions. We take the lower value:
\[ \text{MSR} = 18 \text{ mm} \]
- The circular scale reading (CSR) is given as 34 divisions.

### Step 4: Calculate the Total Reading
The total thickness (T) of the glass plate can be calculated using the formula:
\[ T = \text{MSR} + (\text{CSR} \times \text{LC}) + \text{Ze} \]
Substituting the values:
- MSR = 18 mm
- CSR = 34
- LC = 0.01 mm
- Ze = -1.25 mm

Calculating the contribution from the circular scale:
\[ \text{CSR} \times \text{LC} = 34 \times 0.01 \text{ mm} = 0.34 \text{ mm} \]

Now substituting everything into the total reading formula:
\[ T = 18 \text{ mm} + 0.34 \text{ mm} - 1.25 \text{ mm} \]
Calculating:
\[ T = 18 + 0.34 - 1.25 = 18 + 0.34 - 1.25 = 17.09 \text{ mm} \]

### Final Result
The thickness of the glass plate is:
\[ T = 17.09 \text{ mm} \]

### Step 5: Identify the Zero Error
The zero error is:
\[ \text{Zero Error} = -1.25 \text{ mm} \]

### Summary
- Thickness of the glass plate = 17.09 mm
- Zero error = -1.25 mm