Standard +0.3 This is a standard M4 centre of mass question requiring integration with variable density. The setup is straightforward (one-dimensional rod, simple linear density function), and the method is routine: integrate x·λ(x) and divide by total mass. While it requires careful algebraic manipulation, it's a textbook application of the standard formula with no conceptual surprises.
A straight rod \(AB\) of length \(a\) has variable density. At a distance \(x\) from \(A\) its mass per unit length is \(k(a + 2x)\), where \(k\) is a positive constant. Find the distance from \(A\) of the centre of mass of the rod. [5]
A straight rod $AB$ of length $a$ has variable density. At a distance $x$ from $A$ its mass per unit length is $k(a + 2x)$, where $k$ is a positive constant. Find the distance from $A$ of the centre of mass of the rod. [5]
\hfill \mbox{\textit{OCR M4 2006 Q1 [5]}}