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A non-uniform rod of length L has linear mass density lambda = K*x²/L, where K is a constant and x is the distance from one end. Find the position of the center of mass of the rod from the end where x = 0.

1. 3L/4
2. L/8
3. L/K
4. 3L/K

Correct answer: 3L/4

Solution

For a non-uniform rod, the center of mass is found by x_cm = integral(x*lambda*dx) / integral(lambda*dx). With lambda = K*x²/L, numerator = (K/L)*integral(x³ dx) from 0 to L = (K/L)*(L⁴/4) = KL³/4. Denominator = (K/L)*integral(x² dx) from 0 to L = (K/L)*(L³/3) = KL²/3. Therefore x_cm = (KL³/4)/(KL²/3) = 3L/4.

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