Center of mass for a function defined body

[Redoctober]

Homework Statement

The upper side of a uniform plate of thickness w is given by a function y(x) = 4a+x^2/a. Length of this plate on the x-axis is a . Find x coord of the center of mass of this object, with respect to the origin O . Take density of the plate as p

The Attempt at a Solution

Y(x)= 4a+(x^(2))/a

Can be restated as w.dA=w(4a+(x^(2))/a).dx
therefore using
w.dA.p=dm
w(4a+(x^(2))/a).dx.p=dm

distance from O to CoM qouted as S
will be S= 1/M∫x.dm
S=1/M∫x.w(4a+(x^(2))/a).p.dx
S=1/M.w.p∫(4ax+(x^3)/a).dx

therefore finally , integrating from x=0 to x=a

S=1/M.w.p(2a^3+(a^3)/4)
Mass M can be substituted by = p.w.∫1.dA = p.w.∫y(x).dx from 0 to a

Is this is correct ?! :) Thanks in advance

:)

 

[Spinnor]

You have roughly a rectangle of width a and height 4a. The center of mass you get should be close to (a/2,2a)? Did you get an answer close to this?

 

[Redoctober]

I did it for the x distance from the origin , i got 27/52*a which is about o.519a i.e nearly 1/2a

 

[ehild]

I did it for the x distance from the origin , i got 27/52*a which is about o.519a i.e nearly 1/2a

it is correct.

ehild

 

[asteriatic]

Yes, this is a correct approach to finding the center of mass for a function-defined body. Your use of integrals and substitution is appropriate and your final result looks correct. However, it would be helpful if you could label your equations and steps more clearly for the reader to follow. Additionally, be sure to include units in your final answer to indicate that it is a distance.