Unit And Dimensions

Practice Questions:

Question 101: easy

A screw gauge gives the following readings when used to measure the diameter of a wire
Main scale reading: \(0 \text{mm}\)
Circular scale reading: \(52 \text{divisions}\)
Given that \(1 \text{mm}\) on main scale corresponds to \(100 \text{divisions}\) on the circular scale. The diameter of the wire from the above data is:

[NEET 2020]

1. \(0.026 text{cm}\)

2. \(0.26 text{cm}\)

3. \(0.052 text{cm}\)

4. \(0.52 text{cm}\)

View Answer

Pitch = \(1 \text{mm}\), Number of divisions = \(100\). Least Count (LC) = Pitch / Number of divisions = \(1 \text{mm} / 100 = 0.01 \text{mm}\). Total reading = MSR + (CSR \(\times\) LC) = \(0 \text{mm} + (52 \times 0.01 \text{mm}) = 0.52 \text{mm}\). Convert to cm: \(0.52 \text{mm} = 0.052 \text{cm}\).

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Question 102: easy

A screw gauge has least count of \(0.01 \text{mm}\) and there are \(50 \text{divisions}\) in its circular scale. The pitch of the screw gauge is:

[NEET 2020]

1. \(0.25 \text{mm}\)

2. \(0.5 \text{mm}\)

3. \(1.0 \text{mm}\)

4. \(0.01 \text{mm}\)

View Answer

Least Count (LC) = \(0.01 \text{mm}\), Number of divisions on circular scale = \(50\). The formula for LC is: LC = Pitch / Number of divisions. Therefore, Pitch = LC \(\times\) Number of divisions = \(0.01 \text{mm} \times 50 = 0.5 \text{mm}\).

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Question 103: easy

A student measured the diameter of a small steel ball using a screw gauge of least count \(0.001 \text{cm}\). The main scale reading is \(5 \text{mm}\) and zero of circular scale division coincides with \(25 \text{divisions}\) above the reference level. If screw gauge has a zero error of \(-0.004 \text{cm}\), the correct diameter of the ball is :

[NEET 2018]

1. \(0.053 \text{cm}\)

2. \(0.525 \text{cm}\)

3. \(0.521 \text{cm}\)

4. \(0.529 \text{cm}\)

View Answer

Least Count (LC) = \(0.001 \text{cm}\). Main Scale Reading (MSR) = \(5 \text{mm} = 0.5 \text{cm}\). Circular Scale Reading (CSR) = \(25 \text{divisions}\). Observed Reading = MSR + (CSR \(\times\) LC) = \(0.5 \text{cm} + (25 \times 0.001 \text{cm}) = 0.525 \text{cm}\). Correct Reading = Observed Reading - Zero Error = \(0.525 \text{cm} - (-0.004 \text{cm}) = 0.525 \text{cm} + 0.004 \text{cm} = 0.529 \text{cm}\).

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Question 104: moderate

The mechanical quantity, which has dimensions of reciprocal of mass (\(\text{M}^{-1}\)) is

1. Torque

2. Gravitational constant

3. Angular momentum

4. Coefficient of thermal conductivity

View Answer

The dimensions of the Gravitational constant \(G\) are \([\text{M}^{-1}\text{L}^3\text{T}^{-2}]\). Thus, it has the dimensions of reciprocal of mass.

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Question 105: moderate

The diameter of a spherical bob, when measured with vernier callipers yielded the following values: 3.33 cm, 3.32 cm, 3.34 cm, 3.33 cm and 3.32 cm. The mean diameter to appropriate significant figures is:

1. 3.33 cm

2. 3.32 cm

3. 3.328 cm

4. 3.3 cm

View Answer

The mean of the values is \[\frac{3.33 + 3.32 + 3.34 + 3.33 + 3.32}{5} = 3.328 \text{cm}\]. Since the measurements are up to two decimal places, the rounded final result should also have two decimal places: 3.33 cm.

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Question 106: easy

Which of the following statement is not true?

1. Pressure is a vector quality

2. Relative density is a scalar quantity

3. Coefficient of viscosity is a scalar quantity

4. Surface tension is a scalar quantity

View Answer

Pressure is defined as force per unit area, where the force is perpendicular to the surface. Since it acts equally in all directions, pressure is a scalar quantity.

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Question 107: easy

Following observations were taken with a vernier calliper while measuring the length of a cylinder: \(3.29 \text{cm}\), \(3.28 \text{cm}\), \(3.29 \text{cm}\), \(3.31 \text{cm}\), \(3.28 \text{cm}\). The most accurate length of the cylinder will be

1. \(3.28 \text{cm}\)

2. \(3.29 \text{cm}\)

3. \(3.32 \text{cm}\)

4. \(3.21 \text{cm}\)

View Answer

The most accurate value is the mean value of the observations. Mean length = \[\frac{3.29 + 3.28 + 3.29 + 3.31 + 3.28}{5} = \frac{16.45}{5} = 3.29 \text{cm}\].

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Question 108: moderate

Consider the given statements:

Statement A: Measured value by any measuring instrument has some error.

Statement B: A measurement can have more accuracy and less precision and vice-versa.

Statement C: The magnitude of the difference between the true value of the quantity and the individual measured value is called relative error of measurement.

Choose the correct option.

1. Only statement A is true

2. Only statement B is true

3. Both statements A and C are true

4. Both statements A and B are true

View Answer

Statements A and B are correct properties of measurements. Statement C is incorrect because the described quantity is the absolute error, not the relative error.

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Question 109: easy

Choose the quantities that are dimensionless among the following.

1. \(\frac{\text{Torque}}{\text{Force}}\)

2. \(\frac{\text{Torque}}{\text{Work}}\)

3. \(\frac{\text{Work}}{\text{Energy}}\)

4. Both (2) and (3)

View Answer

Torque, work, and energy all have the same dimensions \([ML^2T^{-2}]\). Hence, \(\frac{\text{Torque}}{\text{Work}}\) and \(\frac{\text{Work}}{\text{Energy}}\) are both dimensionless, while \(\frac{\text{Torque}}{\text{Force}}\) has the dimension of length.

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Question 110: easy

Two physical quantities \(P\) and \(Q\) have different dimensions. The mathematical operation that is not possible among the following are

1. \(\frac{P}{Q}\)

2. \(PQ\)

3. \(P + Q\)

4. Both (1) and (2)

View Answer

According to the principle of homogeneity of dimensions, only quantities with identical dimensions can be added or subtracted. Multiplication and division of different physical quantities are allowed.

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