Find the centre of mass of semicircular ring if surface
Answers
Firstly, the result for y¯ = 2r/π might have been given earlier to the Exercise 5/5 not shown here,for a semi-circular arc, like a wire and not the full area.
It is calculated as y¯ =∫yds∫ds=∫r⋅rsinθdθπr = 2rπ
Secondly, when a thin ring is considered, differential area of a semicircular ring is its arc length multiplied by thickness.
Area of circle = πr2 is already an advanced result from integration for full circle.
In methods of differential calculus we have a clear meaning for differential quantities.When area dA and thin radial slice dr are differentials we are allowed to treat area of ring as that of a thin "curved rectangle". When a thin annular semi circle is considered, differentials only are multiplied.
dA=2πrdr comes at first for thin curved rectangle/ ring and then only comes A=πr2 after performing integration for the full arc.