Question

a 50m long chain hangs vertically from a cunlinder attached to a winch. Assume there is no friction in the system and that the chain has a density of 10kg/m. how much work is required to wind the chain into the cylinder if a 50kg block is attached to the end of the chain?

Answer 1

Answer:

147000 J

Step-by-step explanation:

We are given that

Length of chain=L=50 m

Density of chain=$\rho=10kg/m^3$

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=$\int_{a}^{b}F(y)dy$

We have F(y)=$\rho g(50-y)dy$

a=0 and  b=50

$g=9.8m/s^2$

Using the formula

Work done=$w_1=10\times 9.8\int_{0}^{50}(50-y)dy$

Where Length of chain is (50-y) has to be lifted.

Work done=$w_1=10\times 9.8[50y-\frac{y^2}{2}]^{50}_{0}$

By using the formula $\int x^ndx=\frac{x^{n+1}}{n+1}+C$

Work done=$w_1=10\times 9.8\times (50(50)-\frac{(50)^2}{2})=98\times (2500-1250)=122500 J$

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

$w_2=\int_{0}^{50}mgdy$

We have m=50 kg

$w_2=\int_{0}^{50}50\times 9.8 dy=490[y]^{50}_{0}=490\times 50=24500 J$

The work done required  to wind the chain into the cylinder if a 50 kg block is attached to the end of the chain=$w_1+w_2=122500+24500=147000 J$