Dimensions

Practice Questions:

Question 11: easy

In the given expression of force \(F = a log \left(\frac{b}{x}\right)\) where \(x\) is displacement, the dimensions of \(a\) and \(b\) respectively are

1. \([a] = [MLT^{-2}], [b] = [L]\)

2. \([a] = [M^{-1}], [b] = [L]\)

3. \([a] = [ML^{-1}], [b] = [MLT^{-1}]\)

4. \([a] = [MLT^{-2}], [b] = [ML^2]\)

View Answer

The argument of the logarithm must be dimensionless, so \([b] = [x] = [L]\). Since the logarithm term is dimensionless, \([a] = [F] = [MLT^{-2}]\).

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Question 12: easy

The dimensional formula for impulse is

1. \[MLT^{–1}\]

2. \[M^{–1}LT\]

3. \[M^{–1}LT^{–1}\]

4. \[ML^{–1}T^{–1}\]

View Answer

Impulse is given by \(I = F \cdot t = [MLT^{-2}][T] = [MLT^{-1}]\).

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Question 13: easy

Assertion (A): Only like quantities can be added or subtracted from each other.

Reason (R): Velocity can be subtracted from the velocity gradient.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

By the principle of homogeneity, only physical quantities with identical dimensions can be added or subtracted. Velocity and velocity gradient have different dimensions, hence they cannot be subtracted.

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Question 14: easy

Assertion (A): If a physical quantity has a unit it must have dimension.

Reason (R): There may exist a physical quantity which has dimension but no unit.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

An angle has a unit (radian) but is dimensionless, so A is false. Any physical quantity that possesses dimensions must have a unit, so R is also false.

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Question 15: easy

Assertion (A): When we change the unit of measurement of a quantity, its numerical value changes.

Reason (R): Smaller the unit of measurement smaller is its numerical value.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

The physical magnitude is invariant, expressed as \(n u = \text{constant}\). Therefore, numerical value is inversely proportional to the unit size. A smaller unit leads to a larger numerical value, making R false.

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Question 16: easy

Assertion (A): When an algebraic equation has been derived, it is advisable to check it for dimensional consistency.

Reason (R): This guarantees that the equation is correct.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

A dimensionally consistent equation is not guaranteed to be physically correct, as dimensionless constants cannot be verified through dimensional analysis. Thus, R is false.

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Question 17: easy

Assertion (A): Pressure and energy density have same units in SI.

Reason (R): Dimensions of energy density and pressure are same.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

Pressure has dimensions of \([M L^{-1} T^{-2}]\). Energy density (energy per unit volume) also simplifies to \([M L^{-1} T^{-2}]\). Since their dimensions are identical, they share the same SI units.

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Question 18: easy

Assertion (A): The dimensions of base (fundamental) quantity in other base quantities is always zero.

Reason (R): All derived quantities may be represented dimensionally in terms of fundamental quantities.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

Base quantities are mutually independent and cannot be defined in terms of each other, making the exponent of one base quantity in another zero. Both statements are true, but R is not the explanation of A.

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Question 19: easy

Assertion (A): A unitless quantity never has a non-zero dimension.

Reason (R): A dimensionless quantity never has a unit.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

A quantity without a unit is always dimensionless, so A is true. However, a dimensionless quantity can have a unit (for example, plane angle has the unit radian), making R false.

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Question 20: easy

Assertion (A): Light year and wavelength have same dimensions.

Reason (R): Light year represent time while wavelength represent distance.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

Both light year and wavelength represent distance and have the dimension \([L]\). Since light year represents distance and not time, R is false.

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