Answer:
half the rop to the top of the building = W = 423.75
top of the building = W = 423.75
Step-by-step explanation:
a) How much work W is done in pulling the rope to the top of the building?
When a length of rope xi has been pulled to the top of the building, then the force required to move the remaining rope is 1/2(g(60− xi)), the weight of the rope.
The work to pull an additional small length of rope, ∆x, to the top of the building is 1 /2 ((60 − xi)∆x).
Note that the relationship between xi and ∆x is that for n constant lengths of rope equal to ∆x, then xi = i∆x. The Riemann sum for the work is:
W= ∑ n,i = 1 (1/2(60 - xi). ∆x) = ∑ n,i = 1 (1/2(60 - i∆x). ∆x)
Note that xi = i∆x will not limit to zero as n → ∞. The corresponding integral is:
W = [tex]\int\limits^3_0 {0. 1/2 (60 - x)} \, dx[/tex]
W = [tex]1/2. [60x - 1/2. x^2]^3_0[/tex]
W = 423.75
d) How much work W is done in pulling half the rope to the top of the building?
This requires almost the same Riemann sum except the upper limit on the sum changes:
W= ∑ n/2,i = 1 (1/2(60 - xi). ∆x) = ∑ n,i = 1 (1/2(60 - i∆x). ∆x)
The corresponding integral is:
W = [tex]\int\limits^{1.5}_0 {5. 1/2 (60 - x)} \, dx[/tex]
W = [tex]1/2. [60x - 1/2. x^2]^{1.5}_0 . 5[/tex]
W = 423.75
