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Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
हिन्दीA non-uniform thin rod of length \( L \) lies along the \( x \)-axis with one end at the origin \( x = 0 \). Its linear mass density varies as \( \lambda(x) = \lambda_0 (1 + \frac{x}{L}) \), where \( \lambda_0 \) is a positive constant. The distance of the centre of mass of the rod from the origin is:
Correct answer · Option 4
\( \frac{5}{9}L \)
Explanation
\( M = \int_0^L \lambda_0(1 + x/L)dx = \lambda_0(L + L/2) = \frac{3}{2}\lambda_0 L \). \( \int_0^L x\lambda dx = \lambda_0 \int_0^L (x + x^2/L)dx = \lambda_0(L^2/2 + L^2/3) = \frac{5}{6}\lambda_0 L^2 \). Thus \( X_{\text{cm}} = \frac{\frac{5}{6}\lambda_0 L^2}{\frac{3}{2}\lambda_0 L} = \frac{5}{9}L \).
