(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?
To calculate the work performed in lifting the chain, we can use the formula:
Work = Force × Distance
In this case, we need to calculate the force required to lift the chain and the distance over which the force is applied. Let's break down the problem step by step.
(i) To find the work performed in lifting the entire chain, we need to calculate the force required and the distance traveled.
First, let's find the mass of the chain. Since the mass-density is given as 3 kg/m, the mass per unit length of the chain is 3 kg/m × 20 m = 60 kg.
Next, we can find the force required to lift the chain. The force required to lift an object is equal to its weight, which is given by the formula:
Force = Mass × Gravity
Where gravity is approximately 9.8 m/s².
Force = 60 kg × 9.8 m/s² = 588 N
Now, let's find the distance over which the force is applied. In this case, it is the length of the chain, which is 20 m.
Using the formula for work, we have:
Work = Force × Distance
Work = 588 N × 20 m = 11760 Joules
Therefore, the work performed in lifting the chain so that it is fully extended is 11760 Joules.
(ii) To find the work performed in lifting 1/4 of the chain, we can apply the same process with adjusted values.
First, calculate the mass of 1/4 of the chain. Since the chain's length is 20 m, 1/4 of the chain would be 5 m.
Mass of 1/4 of the chain = 3 kg/m × 5 m = 15 kg
Next, calculate the force required to lift 1/4 of the chain:
Force = Mass × Gravity
Force = 15 kg × 9.8 m/s² = 147 N
The distance over which the force is applied is 5 m.
Using the formula for work, we have:
Work = Force × Distance
Work = 147 N × 5 m = 735 Joules
Therefore, the work performed in lifting 1/4 of the chain is 735 Joules.
To calculate the work performed in lifting the chain, we need to consider the gravitational potential energy. The work done is equal to the change in potential energy.
(i) To find the work performed in lifting the entire chain:
Given:
- Length of chain (L) = 20 m
- Mass density (ρ) = 3 kg/m
Step 1: Find the total mass of the chain.
The total mass (m) of the chain can be calculated by multiplying the length of the chain by the mass density:
m = ρ * L
= 3 kg/m * 20 m
= 60 kg
Step 2: Find the initial and final heights of the chain.
In the initial state, the chain is coiled on the ground. Therefore, the initial height (h_initial) is 0.
In the final state, one end of the chain touches the ground. Therefore, the final height (h_final) is 20 m.
Step 3: Calculate the work done.
The work done (W) can be calculated using the formula:
W = m * g * (h_final - h_initial)
= m * g * (20 m - 0 m)
= 60 kg * 9.8 m/s^2 * 20 m
= 11,760 J
Therefore, the work performed in lifting the chain so that it is fully extended is 11,760 Joules.
(ii) To find the work performed in lifting 1/4 of the chain:
Step 1: Find the mass of 1/4 of the chain.
The mass (m_quarter) of 1/4 of the chain can be calculated by dividing the total mass of the chain by 4:
m_quarter = m / 4
= 60 kg / 4
= 15 kg
Step 2: Find the initial and final heights.
In this case, the initial height is still 0, and the final height is 20 m.
Step 3: Calculate the work done.
Using the same formula as above:
W_quarter = m_quarter * g * (h_final - h_initial)
= 15 kg * 9.8 m/s^2 * 20 m
= 2,940 J
Therefore, the work performed to lift 1/4 of the chain is 2,940 Joules.