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if each cm of main scale is divided in q...

if each cm of main scale is divided in q equal parts and p vernier scale divisions coincide with (p-1) main scale divisions then

A

1 vernier scale division `=1/q((p-1)/p) cm `

B

1 Vernier scale division `=q/p` cm

C

Vernier constant `=(q-1)/(pq)` cm

D

Vernier constant `=1/(pq)` cm

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will derive the formulas for the Vernier scale division (VSD) and the Vernier constant (VC) based on the given information. ### Step 1: Determine the length of one main scale division (MSD) Given that each centimeter of the main scale is divided into \( Q \) equal parts, the length of one main scale division (MSD) is: \[ \text{Length of 1 MSD} = \frac{1 \text{ cm}}{Q} = \frac{1}{Q} \text{ cm} \] **Hint:** Remember that the total length of the main scale is divided into \( Q \) equal parts. ### Step 2: Relate the Vernier scale divisions to the main scale divisions It is given that \( P \) Vernier scale divisions (VSD) coincide with \( P - 1 \) main scale divisions (MSD). This means: \[ P \text{ VSD} = (P - 1) \text{ MSD} \] Substituting the length of one MSD: \[ P \text{ VSD} = (P - 1) \left(\frac{1}{Q} \text{ cm}\right) \] **Hint:** Use the relationship between the number of divisions and their lengths to set up the equation. ### Step 3: Solve for the length of one Vernier scale division (VSD) From the equation above, we can express the length of one Vernier scale division (VSD): \[ \text{Length of 1 VSD} = \frac{(P - 1)}{P} \cdot \frac{1}{Q} \text{ cm} \] This simplifies to: \[ \text{Length of 1 VSD} = \frac{P - 1}{PQ} \text{ cm} \] **Hint:** Isolate the VSD in the equation to find its length. ### Step 4: Calculate the Vernier constant (VC) The Vernier constant is defined as: \[ \text{Vernier Constant (VC)} = 1 \text{ MSD} - 1 \text{ VSD} \] Substituting the values we have: \[ \text{VC} = \frac{1}{Q} \text{ cm} - \frac{P - 1}{PQ} \text{ cm} \] Finding a common denominator: \[ \text{VC} = \frac{P}{PQ} - \frac{P - 1}{PQ} = \frac{P - (P - 1)}{PQ} = \frac{1}{PQ} \text{ cm} \] **Hint:** Remember that the Vernier constant is the difference between one main scale division and one Vernier scale division. ### Final Results 1. **Length of one Vernier scale division (VSD)**: \[ \text{Length of 1 VSD} = \frac{P - 1}{PQ} \text{ cm} \] 2. **Vernier constant (VC)**: \[ \text{VC} = \frac{1}{PQ} \text{ cm} \]
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