Question

Assuming that the mass 'M' of the largest stone that can be moved by a flowing river depends upon velocity'v',density of water(p) and on acceleration due to gravity 'g'.show that mass'M' varies with the sixth power of the velocity of flow.

Answer 1

M -> V, d (density) and g

Let M (proportional to) V^ad^bg^c

Now Hence M = k V^ad^bg^c where k is constant of proportionality

[M] = [V]^a[d]^b[g]^c

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

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

Comparing the powers,

b = 1

a - 3b + c = 0

Hence,

a - 3(1) + c = 0

a + c = 3 ...........1

-a - 2c = 0 ............2

Solving equations 1 and 2,

a + c = 3

-a - 2c = 0

---------------

-c = 3

Hence, c = - 3.

Substituting c = -3 in equation 1, we get a = 6.

Thus, M = k V⁶dg

hence, it can be said that M varies with 6th power of velocity.

OR

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 V⁶dg

∴ 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)