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))`

Answer 1

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.