Question

Mass is non-uniformly distributed over the rod of length $l$ . Its linear mass density varies linearly with length as $\lambda =kx^{2}$ . The position of centre of mass (from lighter end) is given by-

• A. $2 l / 5$
• B. $3 / l$
• C. $3 l / 4$
• D. $2 l / 3$

Answer 1

Answer: (C)

$X_{c m}=\frac{\displaystyle \int x d m}{\displaystyle \int d m}$
$X_{c m}=\frac{\displaystyle \int x \lambda d x}{\displaystyle \int \lambda d x}$
$X_{c m}=\frac{\displaystyle \int _{0}^{1} x \left(k x^{2} d x\right)}{\displaystyle \int _{0}^{1} k x^{2} d x}$
$X_{c m}=\frac{l^{4} / 4}{l^{3} / 3}=\frac{3}{4}l$