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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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