Q.(a) Explain the principle of homogenity of dimensions and derive an expression for the force F acting on a body moving in a circular path depending on the mass of the body (m), velocity
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Start your 14-day free trial to unlock the full solution →The principle of homogeneity of dimensions requires every term in a physical equation to carry the same dimensions; applying it to F depending on m, v, r yields F = m v^2 / r (with k = 1).
This question offers an OR alternative (Bernoulli's Theorem); this solution answers the primary part (a) as instructed.
PRINCIPLE OF HOMOGENEITY OF DIMENSIONS: This principle states that in any valid physical equation, the dimensions of every term on both sides of the equation must be exactly the same (i.e., dimensionally homogeneous/consistent). Only quantities of the same dimensions can be meaningfully added, subtracted, or equated. This principle is the basis of the method of dimensional analysis, used both to check the correctness of an equation and to derive the FORM of a relationship (up to an undetermined dimensionless constant) between physical quantities.
DERIVATION -- Force on a body moving in a circular path:
Suppose the centripetal force F acting on a body of mass m, moving with speed v, along a circular path of radius r, depends on these three quantities as a power-law relationship:
F = k * m^a * v^b * r^c
where k is a dimensionless constant, and a, b, c are unknown powers to be found by dimensional analysis.
Writing the dimensional formula of each quantity:
[F] = [M L T^-2]
[m] = [M]
[v] = [L T^-1]
[r] = [L]
Substituting into the assumed relation:
[M L T^-2] = [M]^a [L T^-1]^b [L]^c = M^a L^(b+c) T^(-b)
Equating powers of M, L, T on both sides:
Power of M: a = 1 …
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