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  4. The centre of gravity (centre of mass) of a rod (of length L) whose linear mass density varies as the square of the distance from one end is at

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Q. The centre of gravity (centre of mass) of a rod (of length $L$) whose linear mass density varies as the square of the distance from one end is at

UPSEEUPSEE 2016

Solution:

Let $A$ be the area of cross section of the rod.
Then, $\bar{x}=\frac{\int_{0}^{L} x \cdot k x^{2} a d x}{\int_{0}^{L} k x^{2} \cdot a d x}=\frac{\int_{0}^{L} x^{3} d x}{\int_{0}^{L} x^{2} d x}=\frac{\frac{L^{4}}{4}}{\frac{L^{3}}{3}}=\frac{3}{4} L$