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CBSE Class 11-science Answered

What is the centre of mass of parabolic area
Asked by rushabhjain.avv | 23 Oct, 2019, 07:18: PM
Expert Answer
Figure shows a lamina of parabolic shape of length L from apex.
 
Let the parobolic curve equation be y2 = k x and apex coincides with origin
 
Let us consider a thin strip of width dx at a distance x from origin.
 
Area of this strip = 2 (kx)1/2 dx ;
If ρ is uniform density ( mass per unit area) of plate then its mass = 2 ρ(kx)1/2 dx
 
By symmetry y-coordinate of centre mass is on x-axis i.e. y = 0 ;
 
x-coordinate cx of Centre of mass of given lamina of parabolic shape  is given by

begin mathsize 12px style c x space equals space 1 over M integral subscript 0 superscript L x space 2 square root of k x end root space space rho space d x end style.........................(1)
Mass of whole lamina is given by

begin mathsize 12px style M space equals space integral subscript 0 superscript L 2 space square root of k space x end root space rho space d x end style ...........................(2)
Hence x-coordinate cx of Centre of mass of given lamina of parabolic shape  is given by
 
begin mathsize 12px style c x space equals space fraction numerator integral subscript 0 superscript L x space 2 square root of k x end root space space rho space d x over denominator integral subscript 0 superscript L space 2 square root of k x end root space space rho space d x end fraction space equals space 3 over 5 L end style
Answered by Thiyagarajan K | 23 Oct, 2019, 08:58: PM
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