Question 11:
moderate
Which pair have not equal dimensions?
[2000]
Force has dimensions \(MLT^{-2}\) and impulse has dimensions \(MLT^{-1}\). These are not equal.
Question 12:
moderate
The dimensions of impulse are equal to that of:
[1996]
Impulse is defined as change in momentum. Therefore, their dimensions are equal, \(MLT^{-1}\).
Question 13:
moderate
The dimensional formula of permeability of free space \(\mu_0\) is
[1991]
From Ampere's law, the dimensions of permeability of free space \(\mu_0\) are derived as \(MLT^{-2}A^{-2}\).
Question 14:
moderate
According to Newton, the viscous force acting between liquid layers of area \(A\) and velocity gradient \( \Delta v / \Delta z \) is given by \(F = -\eta A (\Delta v / \Delta z)\), where \(\eta\) is constant called coefficient of viscosity. The dimensional formula of \(\eta\) is:
[1990]
From the formula \(F = -\eta A (\Delta v / \Delta z)\), the dimension of \(\eta\) is \(F / (A \cdot (\Delta v / \Delta z)) = [MLT^{-2}] / ([L^2] \cdot [T^{-1}]) = [ML^{-1}T^{-1}]\).
Question 15:
moderate
Which of the following quantities, which one has dimensions different from the remaining three?
[1989]
Energy per unit volume, force per unit area, and product of voltage and charge per unit volume all have dimensions \(ML^{-1}T^{-2}\). Angular momentum has dimensions \(ML^2T^{-1}\), which is different.
Question 16:
moderate
If force \([F]\), acceleration \([A]\) and time \([T]\) are chosen as the fundamental physical quantities. Find the dimensions of energy.
[2021]
We know \(E = ML^2T^{-2}\), \(F = MLT^{-2}\), \(A = LT^{-2}\), \(T = T\). From \(F = MA\), \(M = F/A\). Substitute \(M\) into \(E\): \(E = (F/A)L^2T^{-2}\). Also, \(L = AT^2\). So, \(E = (F/A)(AT^2)^2T^{-2} = (F/A) A^2 T^4 T^{-2} = F A T^2\).
Question 17:
moderate
Dimensions of resistance in an electrical circuit, in terms of dimension of mass (M), of length (L), of time (T) and of current (I), would be
[2007]
Resistance \(R = V/I = W/(QI)\). (W) is work done, (Q) is charge. \(W = [ML^2T^{-2}]\), \(Q = [IT]\). So, \(R = [ML^2T^{-2}] / ([IT][I]) = [ML^2T^{-3}I^{-2}]\).
Question 18:
moderate
The velocity (v) of a particle at time (t) is given by \[v = at + \frac{b}{t+c}\] where (a), (b) and (c) are constants. The dimensions of (a), (b) and (c) are respectively:
[2006]
From dimensional homogeneity: ([c] = [t] = [T]). ([at] = [v]) so ([a] = [v]/[t] = [LT^{-1}]/[T] = [LT^{-2}]). ([b/(t+c)] = [v]) so ([b] = [v][t] = [LT^{-1}][T] = [L]).
Question 19:
moderate
The ratio of the dimensions of Planck’s constant and that of the moment of inertia is the dimension of:
[2005]
Planck's constant (h) has dimensions of angular momentum, \([ML^2T^{-1}]\). Moment of inertia (I) has dimensions \([ML^2]\). The ratio \(h/I = [ML^2T^{-1}]/[ML^2] = [T^{-1}]\). \([T^{-1}]\) is the dimension of frequency.
Question 20:
moderate
An equation is given here \[\left(P + \frac{a}{V^2}\right) = b\frac{\theta}{V}\] where P = Pressure, V = Volume and \(\theta =\) Absolute temperature. If (a) and (b) are constants, the dimensions of (a) will be:
From dimensional homogeneity, \([a/V^2] = [P]\). \([a] = [P][V^2]\). Pressure \(P = [ML^{-1}T^{-2}]\), Volume \(V = [L^3]\). So, \([a] = [ML^{-1}T^{-2}][L^3]^2 = [ML^{-1}T^{-2}L^6] = [ML^5T^{-2}]\).