(i) A 20m chain with a mass-density of 3kg/m (coiled on the ground). How

much work is performed lifting the chain so that it is fully extended (and
one end touches the ground)?

(ii) How much work is performed to lift 1/4 of the chain?

This sounds like a physics problem:

Work done = increase in potential energy (PE)
Let
PE at ground = 0
After lifting, the centre of gravity of the chain is at h=10 m above ground, so
potential energy
=mgh
=(3 kg/m * 20m)*9.8 m/s²*10m
= 5880 joules

Using calculus,

Let
ρ = length density = 3 kg/m
x=distance of top of chain from ground in metres
then
length of chain off ground = x
mass of chain off ground = ρx
weight of chain off ground = ρxg
lifting the chain through a distnce dx does work of
ρxg*dx
Total work done through a length of 0 to L metres
=∫ρx;gdx [0,L]
=14.7L²
For L=20m,
Work done = 14.7(20²)=5880 joules
as before.

(i) To calculate the work done in lifting the entire chain, we need to consider the potential energy gained by the chain. The potential energy is equal to the work done against gravity.

The chain has a mass-density of 3 kg/m, which means that for every 1 meter of chain, there is a mass of 3 kg. The total mass of the chain, given a length of 20 meters, can be calculated as:

Mass = mass density x length
Mass = 3 kg/m x 20 m
Mass = 60 kg

The height to lift the chain so that it is fully extended and one end touches the ground is 20 meters.

The formula for calculating potential energy is:
Potential energy = mass x gravitational acceleration x height

Using the values we have:
Potential energy = 60 kg x 9.8 m/s^2 x 20 m
Potential energy = 11,760 joules

Therefore, the work performed in lifting the chain so that it is fully extended and one end touches the ground is 11,760 joules.

(ii) To calculate the work done in lifting 1/4 of the chain, we need to calculate the length and mass for 1/4 of the chain.

The length of 1/4 of the chain can be calculated as:
Length = 1/4 x 20 m
Length = 5 m

The mass of 1/4 of the chain can be calculated as:
Mass = mass density x length
Mass = 3 kg/m x 5 m
Mass = 15 kg

The height to lift 1/4 of the chain so that it is fully extended would also be 5 meters.

Using the same formula for potential energy:
Potential energy = mass x gravitational acceleration x height

Using the values we have:
Potential energy = 15 kg x 9.8 m/s^2 x 5 m
Potential energy = 735 joules

Therefore, the work performed in lifting 1/4 of the chain is 735 joules.

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.