Question

Assuming that the mass M of the largest stone that can be moved by flowing river depends on the velocity of water V density of water p and the acceleration due to gravity G then M is directly proportional to

Answer 1

Assuming that the mass of the largest (M) stone that can be moved by flowing river depends on
velocity (v), the density (ρ), and acceleration due to gravity (g) show that (M) varies directly as the
sixth power of velocity flow (v).

Let M (proportional to) V^a d^b g^c …………. (I)


M = k V^ad^bg^c, where, k is a proportionality constant

[M] = [V]^a[d]^b[g]^c
Writing the dimensions of each physical quantity.

M¹L⁰T⁰ = (L¹T^-1)^a(M¹L^-3)^b(L¹T^-2)^c

M¹L⁰T⁰ = M^b L^{a - 3b + c} T^{-a - 2c}

On Comparing the powers on both sides of the above dimensional equation:

b = 1, a - 3b + c = 0 Hence, a - 3(1) + c = 0
a + c = 3 ------------(i)
-a - 2c = 0 ----------(ii)

On solving (i) and (ii)
a + c = 3 -------------(iii)
-a - 2c = 0------------(iv)
On adding (iii) and (iv)
-c = 3, Hence, c = - 3.

Substituting c = -3 in equation (I),
a = 6.

Thus, M = k V⁶d$g^{-3}$

∴ M is proportional to the 6^th power of V if the mass of the largest (M) stone that can be moved by flowing river depends (i.e., directly proportional to) on velocity (v), the density (ρ), and acceleration due to gravity (g)