How to transform equation 3-20 to equation 3-22a for center of pressure

[goldfish9776]

Homework Statement

pls refer to photo 2 , (circled part) How to transform equation in 3-20 to become equation in 3-22a ?

Homework Equations

The Attempt at a Solution

i have tried in photo 3 , yet can get the same form as in 3-22a . In step 2 , i divided the y_p with Fr

 

[haruspex]

I can see an error in your working. In the left hand term, you substituted F_R=P₀A (wrong), but in the right hand term you correctly used P_cA.
I'll continue to study the text, but it is hard to read in places.

Edit:
I eventually figured out that the denominator in the last term of 3-22a reads $(y_c+P_0/(\rho g \sin(\theta))A$, with which I agree.
Do you get that with the above correction?

 

[goldfish9776]

no , i still gt stucked here

 

[haruspex]

no , i still gt stucked here

Put the denominator in the second term back into the form P_cA for now.
Take the second term in the numerator (the one with y_c²; you missed out the c subscript) and shift it into the first term after the equals sign. In the numerator there, you can extract a factor y_c. The other factor is P_cA, so cancels with the denominator.

 

[goldfish9776]

how to facrorise yc ? there are 3 terms there , but there are only 2 terms contain yc ... and i can't factorise PoA ...

 

[haruspex]

how to facrorise yc ? there are 3 terms there , but there are only 2 terms contain yc ... and i can't factorise PoA ...

You have successfully factored out the y_c. In the other factor, you have two terms each with a denominator P_cA. So you can add together the numerators. I think you'll find those numerators add up to P_cA.

 

[goldfish9776]

You have successfully factored out the y_c. In the other factor, you have two terms each with a denominator P_cA. So you can add together the numerators. I think you'll find those numerators add up to P_cA.

ok , i get my yc now , how to transform the ( (rho)g sin theta ( I xxc) / PcA )into the same form as 3-22a?

 

[haruspex]

ok , i get my yc now , how to transform the ( (rho)g sin theta ( I xxc) / PcA )into the same form as 3-22a?

In that term, expand the P_c in the denominator into P₀+ etc. form, then divide through numerator and denominator by (rho)g sin theta.