KVPY SX SB Exam 2015 Question Paper

Question : 97 of 120

Marks: +1, -0

The distance between the vertex and the center of mass of a uniform solid planar circular segment of angular size θ and radius R is given by -

Solution:

Planar circular segment can be seen as it consist of Arc element

Mass of element

= dm = σ × rθ × dr

centre of mass of Arc element is at

{r\sinθ/2}/{θ/2}

∴ Centre of mass location of segment =

{∑(dm[{r\sinθ\/2} /{θ\/2}])}/{Σdm}

={∫↖{R}↙{0}{{\σrθdr×r\sin(θ\/2)}}/{θ\/2}}/{∫↖{R}↙{0}\σrθdr}

⇒

2[{\sinθ/2}/θ]\;R^3/{3×R^2/2}

⇒

4/3R\;{\sin(θ\/2)}/θ

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