The vernier of a circular scale is divided into `30` divisons, which coincides with `29` main scale divisions. If each main scale division is `(1//2)^@` the least count of the instrument is `(1^@=60')`

Options

A. (a)`0.1`
B. (b)`1'`
C. (c)`10'`
D. (d)`30'`

Explanation

To solve the problem, we need to find the least count of the instrument given the information about the vernier scale and the main scale. Let's break it down step by step.

### Step 1: Understand the relationship between the vernier scale and the main scale.
The problem states that the vernier scale is divided into 30 divisions which coincide with 29 main scale divisions. This can be expressed mathematically as:
\[ 30 \text{ Vernier divisions} = 29 \text{ Main scale divisions} \]

### Step 2: Determine the value of one main scale division.
We are given that each main scale division (MSD) is \( \frac{1}{2} \) degrees. Therefore, we can write:
\[ 1 \text{ MSD} = \frac{1}{2} \text{ degrees} \]

### Step 3: Calculate the value of one vernier scale division.
To find the value of one vernier scale division (VSD), we can use the relationship established in Step 1:
\[ 1 \text{ VSD} = \frac{29}{30} \text{ MSD} \]
Substituting the value of MSD:
\[ 1 \text{ VSD} = \frac{29}{30} \times \frac{1}{2} \text{ degrees} \]
\[ 1 \text{ VSD} = \frac{29}{60} \text{ degrees} \]

### Step 4: Calculate the least count of the instrument.
The least count (LC) of the instrument is given by the formula:
\[ \text{Least Count} = 1 \text{ MSD} - 1 \text{ VSD} \]
Substituting the values we have:
\[ \text{Least Count} = \frac{1}{2} \text{ degrees} - \frac{29}{30} \text{ MSD} \]
To perform this subtraction, we need to express \( \frac{1}{2} \) in terms of 30ths:
\[ \frac{1}{2} = \frac{15}{30} \text{ degrees} \]
Now substituting:
\[ \text{Least Count} = \frac{15}{30} - \frac{29}{30} \]
\[ \text{Least Count} = \frac{15 - 29}{30} \]
\[ \text{Least Count} = \frac{-14}{30} \]
This gives us:
\[ \text{Least Count} = \frac{1}{60} \text{ degrees} \]

### Step 5: Convert the least count to minutes.
Since we are given that \( 1 \text{ degree} = 60 \text{ minutes} \), we can convert the least count:
\[ \text{Least Count in minutes} = \frac{1}{60} \text{ degrees} \times 60 \text{ minutes/degree} \]
\[ \text{Least Count in minutes} = 1 \text{ minute} \]

### Final Answer:
The least count of the instrument is **1 minute**.