The time dependence of a physical quantity (p) is given by \(p = p_0 \text{exp } (-\alpha t^2)\), where (alpha) is constant and (t) is the time. The constant \(\alpha\):
[1993]
For \(\text{exp }(-\alpha t^2)\) to be dimensionless, \(\alpha t^2\) must be dimensionless. \([\alpha][t^2] = [M^0L^0T^0]\). Since \([t] = [T]\), \([\alpha][T^2] = [1]\). Thus, \([\alpha] = [T^{-2}]\).
(P) represents radiation pressure, (c) represents speed of light and (S) represents radiation energy striking per unit area per sec. The non-zero integers (x, y, z) such that \(P^x S^y c^z\) is dimensionless are:
[1992]
Dimensions: \(P = [ML^{-1}T^{-2}]\), \(c = [LT^{-1}]\), \(S = [MT^{-3}]\). For \(P^x S^y c^z\) to be dimensionless, powers of M, L, T must be zero. \(M: x+y=0\). \(L: -x+z=0\). \(T: -2x-3y-z=0\). Solving gives \(y=-x\) and \(z=x\). Taking \(x=1\) yields \(y=-1\), \(z=1\).
The frequency of vibration (f) of a mass (m) suspended from a spring of spring constant (k) is given by a relation \(f = a.m^x k^y\), where (a) is a dimensionless constant. The values of (x) and (y) are:
[1990]
Frequency \(f = [T^{-1}]\). Mass (m = [M]). Spring constant \(k = [MT^{-2}]\). Comparing dimensions of \(f = m^x k^y\): \([T^{-1}] = [M]^x [MT^{-2}]^y = [M^{x+y} T^{-2y}]\). Solving (x+y=0) and (-2y=-1) gives (y = 1/2) and (x = -1/2).