Answer:
Work done[tex]=\pi g(0.365625k+0.39375)[/tex]
Explanation:
To evaluate the force on each strip first we have to find the mass on each strip
Using x from the top of water Volume of strip is
ΔV=π r² × height
ΔV=π(0.5)²×Δx
Now mass of strip is given by:
Mass=Density × volume
ΔM=(1+kh)×π(0.5)²×Δx
Hence h is measured from surface of water so h=x-0.3 So after substitution the value of mass is:
ΔM=(1+k(x-0.3)×π(0.5)²×Δx)
Now the force due to gravity
ΔF=(1+k(x-0.3)×π(0.5)²×Δx)×g
The work done by water in moving distance x
ΔW=ΔF × x
ΔW=[(1+k(x-0.3)×π(0.5)²×Δx)×g]× x
W=[tex]\int\limits^{1.8}_{0.3}[/tex] [[(1+k(x-0.3)×π(0.5)²×Δx)×g]× x]dx
[tex]=\pi g(0.5)^2\int\limits^{13}_{0.3} {(x-0.3kx+kx^2)} \, dx\\ =\frac{\pi g}{4}(1.62+1.485k)- \frac{\pi g}{4}(0.0450-0.00045k)\\=\pi g(0.39375+0.365625k)\\[/tex]
Hence the work done=[tex]=\pi g(0.365625k+0.39375)[/tex]
