Question

Derive how the centre of mass of semicircular disc comes to be 4R/3π

Answer 1

.Notice, your formula YCM=∫y dmYCM=∫y dm is not correct.

the center of mass is given as

YCM=∫y dm∫dmYCM=∫y dm∫dm

Now, substituting the values y=rsinθy=rsin⁡θ& dm=σrdrdθdm=σrdrdθ, we get

YCM=∫R0∫π0σr2sinθdθ dr∫R0∫π0σrdθ drYCM=∫0R∫0πσr2sin⁡θdθ dr∫0R∫0πσrdθ dr

=∫R0(∫π0sinθdθ)r2 dr∫R0(∫π0dθ)r dr=∫0R(∫0πsin⁡θdθ)r2 dr∫0R(∫0πdθ)r dr

=∫R0(2)r2 dr∫R0(π)r dr=2∫R0r2 drπ∫R0r dr=∫0R(2)r2 dr∫0R(π)r dr=2∫0Rr2 drπ∫0Rr dr

=2[r33]R0π[r22]R0=2[r33]0Rπ[r22]0R

=4R33πR2=4R3π=4R33πR2=4R3π

YCM=4R3