If force \([F]\), acceleration \([A]\) and time \([T]\) are chosen as the fundamental physical quantities. Find the dimensions of energy.
Energy has the dimensions of work, which is \(\text{Force} \times \text{displacement}\). Since displacement has the dimensions of \(text{acceleration} \times \text{time}^2 = [A][T^2]\), the dimensional formula of energy is \([F][A][T^2]\).
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
The argument of the logarithm must be dimensionless, so \([b] = [x] = [L]\). Since the logarithm term is dimensionless, \([a] = [F] = [MLT^{-2}]\).
The dimensional formula for impulse is
Impulse is given by \(I = F \cdot t = [MLT^{-2}][T] = [MLT^{-1}]\).
Assertion (A): In mechanics the method of dimensions can’t be applied to derive formula of a physical quantity which depends on more than three physical quantities.
Reason (R): We can derive relation of a physical quantity with other physical quantities out of which two have same dimensions.
In mechanics, we have only three base dimensions (M, L, T). Thus, we cannot determine more than three independent exponents. If two quantities have the same dimensions, they cannot be resolved independently, making R false.
Assertion (A): Only like quantities can be added or subtracted from each other.
Reason (R): Velocity can be subtracted from the velocity gradient.
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.
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.
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.
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.
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.
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.
A dimensionally consistent equation is not guaranteed to be physically correct, as dimensionless constants cannot be verified through dimensional analysis. Thus, R is false.
Assertion (A): Pressure and energy density have same units in SI.
Reason (R): Dimensions of energy density and pressure are same.
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.
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.
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.