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, \(E = (F/A)(AT^2)^2T^{-2} = (F/A) A^2 T^4 T^{-2} = F A T^2\).
If Force \((F)\), Velocity \((V)\), and Time \((T)\), are taken as fundamental units, then the dimensions of mass are:
[2014]
Given fundamental units: Force \(F = [MLT^{-2}]\), Velocity \(V = [LT^{-1}]\), Time \(T = [T]\). We want to find dimensions of Mass \(M = F^x V^y T^z\). Equating dimensions: \([M] = [MLT^{-2}]^x [LT^{-1}]^y [T]^z = [M^x L^{x+y} T^{-2x-y+z}]\). Comparing powers: \(x=1\), \(x+y=0 ⇒ 1+y=0 ⇒ y=-1\), \(-2x-y+z=0 ⇒ -2(1)-(-1)+z=0⇒ -2+1+z=0 ⇒ z=1\). Thus, mass dimensions are \(FV^{-1}T\).
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, \([a] = [ML^{-1}T^{-2}][L^3]^2 = [ML^{-1}T^{-2}L^6] = [ML^5T^{-2}]\).
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, \(R = [ML^2T^{-2}] / ([IT][I]) = [ML^2T^{-3}I^{-2}]\).
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]).
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.
The mechanical quantity, which has dimensions of reciprocal of mass (\(\text{M}^{-1}\)) is
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.