First slide

Methods to locate centre of mass

Question

The density of a non-uniform rod of length 1m is given by λ(x) = p(1 + qx²) where p and q are constants and 0≤x≤1. The centre of mass of the rod will be at ...........

Moderate

A

$\large \frac{3}{4}\left[ {\frac{{2 + q}}{{3 + q}}} \right]$
B

$\large \frac{4}{3}\left[ {\frac{{2 + q}}{{3 + q}}} \right]$
C

$\large \frac{3}{4}\left[ {\frac{{3 + q}}{{2 + q}}} \right]$
D

$\large \frac 43\left[ {\frac{{3 + q}}{{2 + q}}} \right]$

Solution

$\large {X_{CM}} = \frac{{\int\limits_0^L {dmx} }}{{\int\limits_0^L {dm} }} = \frac{{\int\limits_0^L {\lambda xdx} }}{{\int\limits_0^L {\lambda dx} }} = \frac{3}{4}\left[ {\frac{{2 + q}}{{3 + q}}} \right]$

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