To find the work performed in lifting the chain, we'll need to use the formula:
Work = Force * Distance
In this case, the force required to lift the chain is equal to its weight, and the distance is the height it is lifted. Let's go step by step to find the answer to each question.
(i) To lift the entire chain and fully extend it, we need to lift it to a height of 20 meters. First, we need to determine the total mass of the chain. The mass of the entire chain can be calculated by multiplying the mass density with the length:
Mass = Mass density * Length
Mass = 3 kg/m * 20 m
Mass = 60 kg
The weight of the chain is equal to the mass multiplied by the acceleration due to gravity (9.8 m/s^2):
Weight = Mass * acceleration due to gravity
Weight = 60 kg * 9.8 m/s^2
Weight = 588 N
Now that we have the weight and the distance, we can calculate the work:
Work = Force * Distance
Work = Weight * Height
Work = 588 N * 20 m
Work = 11,760 Joules
Therefore, the work performed in lifting the entire chain is 11,760 Joules.
(ii) To lift only 1/4 of the chain, we need to calculate the length and the mass of that part.
Length of 1/4 chain = 1/4 * 20 m
Length of 1/4 chain = 5 m
Mass of 1/4 chain = Mass density * Length
Mass of 1/4 chain = 3 kg/m * 5 m
Mass of 1/4 chain = 15 kg
To find the weight, we use the same equation as before:
Weight of 1/4 chain = Mass * acceleration due to gravity
Weight of 1/4 chain = 15 kg * 9.8 m/s^2
Weight of 1/4 chain = 147 N
Now, we can calculate the work:
Work = Force * Distance
Work = Weight * Height
Work = 147 N * 5 m
Work = 735 Joules
Therefore, the work performed to lift 1/4 of the chain is 735 Joules.