First slide

Methods to locate centre of mass

Question

The linear density of a thin rod of length 1 m varies as λ = 1 + 2x , where x is the distance from its one end. Distance of centre of mass of rod from this end is :

Difficult

A

5/12 m
B

3/4 m
C

7/12 m
D

2/3 m

Solution

$dm = λdx = (1 + 2 x) dx$
$x_{cm} = \frac{\int_{0}^{1} (d m) x}{\int_{0}^{1} d m} = \frac{\int_{0}^{1} x (1 + 2 x) d x}{\int_{0}^{1} (1 + 2 x) d x} = \frac{7}{12} m$

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