In a screw guage the graduation in the circular scale consists of 100 equal divisions and 10 complete rotation of athe cap is equivalent to 5mm on the main scale . While measuring diameter of a wire with screw gauge a particular observation gives initial circular cap reading as 57 and after completing 3 complete rotation the final circular cap reading as 63 . it is observed that while proceeding from initial reading to final reading the cap reading increases. calculate the diameter of the wire from the above data.

Answer

To calculate the diameter of the wire using the given data from the screw gauge, we can follow these steps:

### Step 1: Determine the distance moved per complete rotation
Given that 10 complete rotations of the cap correspond to 5 mm on the main scale, we can find the distance moved per complete rotation.

\[ \text{Distance per rotation} = \frac{5 \text{ mm}}{10} = 0.5 \text{ mm} \]

### Step 2: Calculate the total distance moved for 3 complete rotations
Since we need to find the total distance moved after 3 complete rotations, we multiply the distance per rotation by the number of rotations.

\[ \text{Distance for 3 rotations} = 3 \times 0.5 \text{ mm} = 1.5 \text{ mm} \]

### Step 3: Calculate the change in circular scale reading
The initial circular scale reading is 57, and the final reading after 3 complete rotations is 63. We can find the change in the circular scale reading:

\[ \text{Change in circular scale reading} = 63 - 57 = 6 \]

### Step 4: Convert the change in circular scale reading to mm
Since the circular scale has 100 equal divisions, we can find the value of each division. The least count (LC) of the screw gauge can be calculated as follows:

\[ \text{Least Count (LC)} = \frac{\text{Distance moved in one complete rotation}}{\text{Number of divisions in circular scale}} = \frac{0.5 \text{ mm}}{100} = 0.005 \text{ mm} \]

Now, we can convert the change in circular scale reading to mm:

\[ \text{Distance from circular scale reading} = \text{Change in reading} \times \text{Least Count} = 6 \times 0.005 \text{ mm} = 0.03 \text{ mm} \]

### Step 5: Calculate the total distance moved
Now, we can find the total distance moved by adding the distance moved due to rotations and the distance moved due to the circular scale reading:

\[ \text{Total distance moved} = \text{Distance for 3 rotations} + \text{Distance from circular scale reading} = 1.5 \text{ mm} + 0.03 \text{ mm} = 1.53 \text{ mm} \]

### Step 6: Conclusion
Thus, the diameter of the wire measured using the screw gauge is:

\[ \text{Diameter of the wire} = 1.53 \text{ mm} \]