JEE Physics Units And Measurements MCQs Set 05

Answer: (c) 2 mm

Explanation:
1. 20 VSD = 19 MSD, so 1 VSD = \(\frac{19}{20}\) MSD
2. Least count = 1 MSD - 1 VSD = \(\frac1{20}\) MSD
3. \(\frac{1}{20}\text{MSD}=0.1\) mm, so 1 MSD = 2 mm

Teacher's Note:
1. Least count of a vernier = value of one main-scale division divided by the number of vernier divisions.

Answer: (c) \(p\) and \(a\)

Explanation:
1. Take logarithms and differentiate: \(\frac{\Delta y}{y}=a\,\frac{\Delta x}{x}\)
2. So the percentage error in \(y\) is \(a\times p\)
3. It depends only on \(p\) and \(a\); the constant \(k\) does not matter

Teacher's Note:
1. In a product or power law, relative errors add (multiplied by the powers). Constant multipliers do not affect relative error.

Answer: (d) \(\Delta L=0.1\) cm, \(\Delta T=0.1\) s

Explanation:
1. \(\frac{\Delta g}{g}=\frac{\Delta L}{L}+2\frac{\Delta T}{T}\)
2. For the same \(L\) and \(T\), the error is smallest when both \(\Delta L\) and \(\Delta T\) are smallest
3. That is the case \(\Delta L=0.1\) cm, \(\Delta T=0.1\) s

Teacher's Note:
1. The period appears squared, so errors in time are doubled in the final result. Both absolute errors must be small to minimise the error in \(g\).

Answer: (c) 0.5 %

Explanation:
1. \(V\propto d^3\)
2. \(\frac{\Delta V}{V}=3\frac{\Delta d}{d}\)
3. \(\frac{\Delta d}{d}=\frac{1.5}{3}=0.5\) %

Teacher's Note:
1. Multiply the percentage error by the power of the quantity. To go back from the volume to the diameter, divide by 3.

Answer: (b) 1.0 \(\pm\) 0.2

Explanation:
1. \(y=2.0-1.0=1.0\)
2. For a difference, the absolute errors add: \(0.1+0.1=0.2\)
3. \(y=1.0\pm0.2\)

Teacher's Note:
1. Absolute errors add in both sums and differences. Never subtract errors.

Answer: (a) Vernier callipers only

Explanation:
1. The metal volume = outer volume - inner volume
2. The vernier callipers can measure the outer diameter and length, the inner diameter and the inner depth (with the depth rod)
3. No other instrument is needed; density = mass / metal volume

Teacher's Note:
1. A vernier calliper has jaws for outer and inner diameters and a depth rod for the depth of a hollow cylinder.

Answer: (c) 5

Explanation:
1. Leading zeros are not significant
2. The digits 6, 9 and the three trailing zeros after the decimal point are significant
3. Total = 5

Teacher's Note:
1. Trailing zeros after a decimal point are significant because they show the precision of the measurement.