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If the linear density (mass per unit length) of a rod of length 3m is proportional to x, where x is the distance from one end of the rod, the distance of the centre of gravity of the rod from this end is:

1. 2.5m
2. 1m
3. 1.5m
4. 2m

Correct answer: 2m

Solution

The linear density is proportional to x, so the mass element is proportional to x dx. The center of gravity is calculated using the formula for the center of mass: \( x_{cm} = \frac{\int x \cdot \lambda(x) dx}{\int \lambda(x) dx} \). Substituting \( \lambda(x) \propto x \), the result simplifies to \( x_{cm} = \frac{2L}{3} \), which gives 2m for L = 3m.

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