Variable density rod or lamina

2 A rod \(A B\) of variable density has length 2 m . At a distance \(x\) metres from \(A\), the rod has mass per unit length ( \(0.7 - 0.3 x ) \mathrm { kg } \mathrm { m } ^ { - 1 }\). Find the distance of the centre of mass of the rod from \(A\).

2 A straight \(\operatorname { rod } A B\) has length \(a\). The rod has variable density, and at a distance \(x\) from \(A\) its mass per unit length is \(k \mathrm { e } ^ { - \frac { x } { a } }\), where \(k\) is a constant. Find, in an exact form, the distance of the centre of mass of the rod from \(A\).

2 A straight \(\operatorname { rod } A B\) has length \(a\). The rod has variable density, and at a distance \(x\) from \(A\) its mass per unit length is given by \(k \left( 4 - \sqrt { \frac { x } { a } } \right)\), where \(k\) is a constant. Find the distance from \(A\) of the centre of mass of the rod.

1. \hspace{0pt} [In this question, you may assume that the centre of mass of a circular arc, radius \(r\), with angle at centre \(2 \alpha\), is a distance \(\frac { r \sin \alpha } { \alpha }\) from the centre.]

\begin{figure}[h] \end{figure} A thin non-uniform metal plate is in the shape of a sector \(O A B\) of a circle with centre \(O\) and radius \(a\). The angle \(A O B = \frac { \pi } { 2 }\), as shown in Figure 5. The plate is modelled as a non-uniform lamina.
The mass per unit area of the lamina, at any point \(P\) of the lamina, is modelled as \(k ( O P ) ^ { 2 }\), where \(k = \frac { 4 \lambda } { \pi a ^ { 4 } }\) and \(\lambda\) is a constant. Using the model,

1. find the mass of the plate in terms of \(\lambda\),
2. find, in terms of \(a\), the distance of the centre of mass of the plate from \(O\).