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: Understand the relationship between rotations and measurement
We know that 10 complete rotations of the screw gauge cap correspond to a movement of 5 mm on the main scale. Therefore, we can find the distance moved by the screw gauge for one complete rotation.

\[ \text{Distance for 1 rotation} = \frac{5 \text{ mm}}{10} = 0.5 \text{ mm} \]

### Step 2: Calculate the total distance moved during the rotations
The problem states that the cap makes 3 complete rotations. Therefore, the total distance moved by the screw gauge during these rotations can be calculated as:

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

### Step 3: Determine the change in circular scale reading
The initial circular scale reading is 57, and after 3 rotations, the final reading is 63. The change in reading can be calculated as:

\[ \text{Change in reading} = \text{Final reading} - \text{Initial reading} = 63 - 57 = 6 \]

### Step 4: Convert the change in circular scale reading to mm
Since the circular scale has 100 equal divisions, each division corresponds to:

\[ \text{Value of each division} = \frac{0.5 \text{ mm}}{100} = 0.005 \text{ mm} \]

Now, we can find the distance corresponding to the change in reading:

\[ \text{Distance for change in reading} = 6 \times 0.005 \text{ mm} = 0.03 \text{ mm} \]

### Step 5: Calculate the total diameter of the wire
The total diameter of the wire can be calculated by adding the distance moved by the screw gauge (from the rotations) and the distance due to the change in circular scale reading:

\[ \text{Diameter of wire} = \text{Distance for 3 rotations} + \text{Distance for change in reading} \]

Substituting the values we calculated:

\[ \text{Diameter of wire} = 1.5 \text{ mm} + 0.03 \text{ mm} = 1.53 \text{ mm} \]

### Final Answer
Thus, the diameter of the wire is **1.53 mm**.

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