A 29-ft chain with mass density 3 lb/ft is initially coiled on the ground. How much work is performed in lifting the chain so that it is fully extended (and one end touches the ground).
bruh
To find the work performed in lifting the chain, we need to calculate the gravitational potential energy gained.
Gravitational potential energy is given by the formula:
PE = mgh
Where:
PE is the potential energy
m is the mass
g is the acceleration due to gravity
h is the height
In this case, we need to consider the mass of each small element of the chain. Since the chain has a mass density of 3 lb/ft, we can consider the chain as a continuous system and integrate the potential energy over the length of the chain.
First, let's find the total mass of the chain. The mass can be calculated by multiplying the mass density by the length of the chain:
m = 3 lb/ft * 29 ft = 87 lb
Next, we need to calculate the height. The height is the distance from each small element of the chain to the ground. Since the chain starts coiled on the ground, each small element has a different height. We can integrate the height from 0 to the length of the chain.
Let's assume the height is denoted by h(x), where x is the position along the length of the chain. At any given point x, the height can be calculated using similar triangles:
h(x) = x * (29 ft / 87 ft) = x / 3 ft
Now, we have all the information we need to calculate the potential energy gained. We can integrate the potential energy over the length of the chain:
PE = ∫[0,29] (m * g * h(x)) dx
Substituting the values:
PE = ∫[0,29] (87 lb * 32.2 ft/s^2 * (x / 3 ft)) dx
Now we can integrate the equation with respect to x:
PE = 87 lb * 32.2 ft/s^2 * ∫[0,29] (x / 3 ft) dx
PE = 87 lb * 32.2 ft/s^2 * (1/3) * ∫[0,29] x dx
PE = 87 lb * 32.2 ft/s^2 * (1/3) * [(1/2) * x^2] |[0,29]
PE = 87 lb * 32.2 ft/s^2 * (1/3) * [(1/2) * (29^2) - (1/2) * (0^2)
PE = 87 lb * 32.2 ft/s^2 * (1/3) * (1/2) * (29^2)
PE ≈ 120775.07 lb * ft^2/s^2
Finally, we have calculated the potential energy gained. This value represents the work performed in lifting the chain. Therefore, the amount of work performed in lifting the chain so that it is fully extended is approximately 120775.07 lb * ft^2/s^2.