A rod of length L is placed along the X-axis between x=0 and x=L . The linear density (mass/length) р of the rod varies with the distance x from the origin as р = a+bx
(A) Find the SI units of a and b.
(B) Find the mass of the rod in terms of a , b and L.
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ANSWER::
ρ = mass/length = a + bx
(a) S.I. unit of 'a' = kg/m
S.I unit of 'b' = kg/m²
i.e. due to the principle of homogeneity of dimensions
(b) Consider a small element of length 'dx' at a distance x from the origin
Therefore , dm = mass of element = ρ dx = (a+b)x
Mass of the road = m = ∫dm = ₐ∫ᵇ (a+bx) dx = ₐ[ax + (bx²/2)]ᵇ = aL + (bL²/2)
Here a = 0 , b = L
Hope it helps!
ANSWER::
ρ = mass/length = a + bx
(a) S.I. unit of 'a' = kg/m
S.I unit of 'b' = kg/m²
i.e. due to the principle of homogeneity of dimensions
(b) Consider a small element of length 'dx' at a distance x from the origin
Therefore , dm = mass of element = ρ dx = (a+b)x
Mass of the road = m = ∫dm = ₐ∫ᵇ (a+bx) dx = ₐ[ax + (bx²/2)]ᵇ = aL + (bL²/2)
Here a = 0 , b = L
Hope it helps!
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BloomingBud:
Why did you deleted my comment? may I know the reason? I didn't use any word. I just put '...' to get the notification from this question.
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