The centre of mass of a non-uniform rod of length L, whose linear mass density lambda = K*x² / L (where K is a constant and x is the distance from one end), is located at

3L/4
L/8
L/K
3L/K

Correct answer: 3L/4

Solution

x_cm = [integral₀^L x*(Kx²/L) dx] / [integral₀^L (Kx²/L) dx]. Numerator = (K/L)*[x⁴/4]₀^L = K*L³/4. Denominator = (K/L)*[x³/3]₀^L = K*L²/3. x_cm = (K*L³/4)/(K*L²/3) = (L/4)*(3) = 3L/4.

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