A rod extends from x = 0 to x = L and has linear mass density λ(x) = kx². If x(CM) = NL/4, find N.

Question

G1 Physics · Accepted Answer

Here dm = kx²dx. Total mass M = ∫₀ᴸ kx²dx = kL³/3. The first moment about the origin is ∫₀ᴸ x dm = ∫₀ᴸ kx³dx = kL⁴/4. Hence x(CM) = (kL⁴/4)/(kL³/3) = 3L/4. Comparing with x(CM) = NL/4 gives N = 3.