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Question
The density of a non-uniform rod of length 1 m is given by ρ(x) = a(1 + bx2) where a and b are constants and 0 ≤ x ≤ 1. The centre of mass of the rod will be at ______.
Options
`(3(2 + b))/(4(3 + b))`
`(4(2 + b))/(3(3 + b))`
`(3(3 + b))/(4(2 + b))`
`(4(3 + b))/(3(2 + b))`
Solution
The density of a non-uniform rod of length 1 m is given by ρ(x) = a(1 + bx2) where a and b are constants and 0 ≤ x ≤ 1. The centre of mass of the rod will be at `underline((3(2 + b))/(4(3 + b)))`.
Explanation:
Density is given as ρ(x) = a(1 + bx2)
Where a and b are constant and 0 ≤ x ≤ 1
Let b → 0, in this case ρ(x) = a = constant
Hence, the centre of mass will be at x = 0.5 m. (middle of the rod)
Putting b = 0 in all the options, only (a) gives 0.5.
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