# Solved Problems , Units & Dimensions

Problem 1: If nth division of main scale coincides with (n+1)th divisions of vernier scale. Given one main scale division is equal to ‘a’ units, find the least count of the vernier.

Solution: (n + 1) division of vernier scale = n division of main scale

∴ one Vernier division = n/(n+1) main scale division

Least count = 1 M.S.D. – 1 V.S.D.

$\large = (1 – \frac{n}{n+1})M.S.D$

$\large = \frac{1}{n+1} M.S.D = \frac{a}{n+1}$

Problem 2: Two consecutive lengths of a resonance column taken with a tuning fork of frequency 480 Hz at 250 °C are 54 cm and 88.4 cm respectively. Find the velocity of sound in air.

Solution: If wavelength =  λ,

λ/2 = (88.4 – 54) cm = 34.4 cm

⇒ λ  = 68.8 cm

∴ c (velocity of sound in air) =  480 m/s ×  0.688 =  330 m/s

Problem 3: If all measurements in an experiment are taken upto same number of significant figures then mention two possible reasons for maximum error.

Solution: The maximum error will be due to (i) measurement, which is least accurate.

(ii) measurement of the quantity which has maximum power in formulas.

Problem 4: The initial and final temperature of water as recorded by an observer are
(40.6 ± 0.2)°C and (78.3 ± 0.3 )°C. Calculate the rise in temperature with proper error limit.

Solution: Let θ1 = 40.6 °C,  Δθ1 = ± 0.2 °C

θ2 = 78.3 °C,  Δθ2 = ± 0.3  °C

⇒ θ = θ2 – θ1 = 78.3 – 40.6 = 37.7 °C

& Δθ = ± (Δθ1 + Δθ2)= ± (0.2 + 0.3) = ± 0.5°C

Hence rise in temperature

= (37.7 ± 0.5)°C

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