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NEET
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Physics
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Experimental Physics

NEET · Physics · Experimental Physics

The main scale of a vernier calliper has m divisions per...

The main scale of a vernier calliper has mm divisions per centimetre. It is given that nn divisions of the vernier scale coincide with (n−1)(n-1) divisions of the main scale. While measuring the length of a cylinder, the zero of the vernier scale lies just after the kthk^{\text{th}} mark of the main scale, and the pthp^{\text{th}} division of the vernier scale coincides with a main scale mark. The measured length of the cylinder is

Options

  1. Ak+pmn cm\dfrac{k+p}{mn} \text{ cm}
  2. Bkn+pmn cm\dfrac{kn+p}{mn} \text{ cm}
  3. Ckn−pmn cm\dfrac{kn-p}{mn} \text{ cm}
  4. Dk+pm cm\dfrac{k+p}{m} \text{ cm}

Correct answer

B. kn+pmn cm\dfrac{kn+p}{mn} \text{ cm}

Step-by-step solution

The length of one main scale division is:
1 MSD=1m cm1 \text{ MSD} = \dfrac{1}{m} \text{ cm}

From the coincidence condition:
n VSD=(n−1) MSDn \text{ VSD} = (n-1) \text{ MSD}
⇒1 VSD=n−1n MSD\Rightarrow 1 \text{ VSD} = \dfrac{n-1}{n} \text{ MSD}

The Least Count (LC) is:
LC=1 MSD−1 VSD=1n MSD=1mn cm\text{LC} = 1 \text{ MSD} - 1 \text{ VSD} = \dfrac{1}{n} \text{ MSD} = \dfrac{1}{mn} \text{ cm}

The total measured length is the sum of the Main Scale Reading (MSR) and the Vernier Scale Reading (VSR):
MSR=k×1 MSD=km cm\text{MSR} = k \times 1 \text{ MSD} = \dfrac{k}{m} \text{ cm}
VSR=p×LC=pmn cm\text{VSR} = p \times \text{LC} = \dfrac{p}{mn} \text{ cm}

Total length=MSR+VSR=km+pmn=kn+pmn cm\text{Total length} = \text{MSR} + \text{VSR} = \dfrac{k}{m} + \dfrac{p}{mn} = \dfrac{kn+p}{mn} \text{ cm}

Answer: kn+pmn cm\dfrac{kn+p}{mn} \text{ cm}
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