A thin rod of length L lies along the x-axis from x = 0 to x = L. Its linear mass density varies as λ(x) = λ₀(1 + x/L). The position of its centre of mass is:
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A thin rod of length L lies along the x-axis from x = 0 to x = L. Its linear mass density varies as λ(x) = λ₀(1 + x/L). The position of its centre of mass is:

G1 Physics · Accepted Answer

Use x(CM) = [∫x dm]/[∫dm], with dm = λ(x)dx. Total mass: M = ∫₀ᴸ λ₀(1 + x/L)dx = λ₀[L + L/2] = 3λ₀L/2. First moment: ∫₀ᴸ xλ₀(1 + x/L)dx = λ₀[∫₀ᴸ xdx + (1/L)∫₀ᴸ x²dx] = λ₀[L²/2 + L²/3] = 5λ₀L²/6. Therefore x(CM) = (5λ₀L²/6)/(3λ₀L/2) = 5L/9.