A metal chain weighs 30 lbs and is 10 feet long. How much work is required to lift one end of the chain to a height of 15 feet (so that the other end of the chain will be at a height of 5 feet)?
To determine the amount of work required to lift the chain, we need to calculate the gravitational potential energy gained. The formula for gravitational potential energy is:
Potential energy (PE) = mass (m) × gravitational acceleration (g) × height (h)
In this case, the mass of the chain is given as 30 lbs. However, since we need to consider the entire length of the chain, we need to convert this mass to a linear mass density.
Linear mass density (λ) = mass (m) / length (L)
The length of the chain is given as 10 feet. Hence:
Linear mass density (λ) = 30 lbs / 10 ft = 3 lbs/ft
Now, we can calculate the mass of the chain for the specific height we need to lift it. Since the chain is 10 feet long and we need to lift one end to a height of 15 feet, the height difference is 15 - 5 = 10 feet.
Mass (m) = linear mass density (λ) × height (h)
Mass (m) = 3 lbs/ft × 10 ft = 30 lbs
Now that we know the mass of the chain for the given height, we can calculate the work required.
Potential energy (PE) = mass (m) × gravitational acceleration (g) × height (h)
Gravitational acceleration (g) is approximately 32.2 ft/s^2.
Potential energy (PE) = 30 lbs × 32.2 ft/s^2 × 10 ft = 9,660 ft-lbs
Therefore, the work required to lift one end of the chain to a height of 15 feet is 9,660 ft-lbs.