Q6. Assuming that the mass m of the largest stone that can be moved by a flowing river depends on the velocity v, the density of water ρ and acceleration due to gravity g. Show that m varies as the sixth power of the velocity of flow
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Answers
Answered by
3
From Given:-
since Mass ∝V
a
d
b
g
c
⇒M=kV
a
d
b
g
c
Equating dimensions.
[M
′
]=k[L
1
T
−1
]
a
[ML
−3
]
b
[L
1
T
−2
]
c
⇒ on comparing:-
b=1,a−3b+c=0
⇒a+c=3......(1)
and −a−2c=0......(2)
on solving:-
c=−3 and a=6
∴M=k
g
3
V
6
d
∴ Mass ∝ sixth Power of Velocity
Hope it helps
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Answered by
0
Answer:
m ∝ v⁶.ρ¹.g⁻³
Explanation:
m ∝ vᵃ.ρᵇ.gˣ
vᵃ = [L¹T⁻¹]ᵃ
ρᵇ = [M¹L⁻³]ᵇ
gˣ = [L¹T⁻²]ˣ
Using dimensional analysis
[M¹L⁰T⁰] ∝ [L¹T⁻¹]ᵃ.[M¹L⁻³]ᵇ. [L¹T⁻²]ˣ
[M¹L⁰T⁰] ∝ [Lᵃ⁻³ᵇ⁺ˣ].[T⁻ᵃ⁻²ˣ].[M]ᵇ
[M¹] ∝ [M]ᵇ
b = 1
[L⁰] ∝[Lᵃ⁻³ᵇ⁺ˣ]
a - 3b + x = 0
a - 3 + x = 0
a + x = 3 -----{i}
[T⁰] ∝ [T⁻ᵃ⁻²ˣ]
-a - 2x = 0 ----{ii}
add {i} and {ii}
a + x - a - 2x = 3
x - 2x = 3
-x = 3
x = -3
a - 3 = 3
a = 6
b = 1 , x = -3 , a = 6
m ∝ vᵃ.ρᵇ.gˣ
m ∝ v⁶.ρ¹.g⁻³
m ∝ v⁶
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