Answer:
147000 J
Step-by-step explanation:
We are given that
Length of chain=L=50 m
Density of chain=[tex]\rho=10kg/m^3[/tex]
We have to find the work done required to wind the chain into the cylinder if a 50 kg block is attached to the end of the chain.
Work done=[tex]\int_{a}^{b}F(y)dy[/tex]
We have F(y)=[tex]\rho g(50-y)dy[/tex]
a=0 and b=50
[tex]g=9.8m/s^2[/tex]
Using the formula
Work done=[tex]w_1=10\times 9.8\int_{0}^{50}(50-y)dy[/tex]
Where Length of chain is (50-y) has to be lifted.
Work done=[tex]w_1=10\times 9.8[50y-\frac{y^2}{2}]^{50}_{0}[/tex]
By using the formula [tex]\int x^ndx=\frac{x^{n+1}}{n+1}+C[/tex]
Work done=[tex]w_1=10\times 9.8\times (50(50)-\frac{(50)^2}{2})=98\times (2500-1250)=122500 J[/tex]
When the chain is weightless then the work done required to lift the block attached to the 50 m long chain
Again using the formula
Where f(y)=mg
[tex]w_2=\int_{0}^{50}mgdy[/tex]
We have m=50 kg
[tex]w_2=\int_{0}^{50}50\times 9.8 dy=490[y]^{50}_{0}=490\times 50=24500 J[/tex]
The work done required to wind the chain into the cylinder if a 50 kg block is attached to the end of the chain=[tex]w_1+w_2=122500+24500=147000 J[/tex]
