Solve the problem.
-A 60-m-long, 9.1-mm-diameter rope hangs free from a ledge. The density of the rope is 70 g/m. How much work is needed to pull the entire rope to the ledge?
A) 4.2g J
B) 9 g J
C) 126,000g J
D) 126 g J

Question

Solve the problem.   
-A 60-m-long, 9.1-mm-diameter rope hangs free from a ledge. The density of the rope is 70 g/m. How much work is needed to pull the entire rope to the ledge?
A) 4.2g J
B) 9 g J
C) 126,000g J
D) 126 g J

Quizplus · Accepted Answer

The work needed to pull the rope to the ledge can be calculated by integrating the force needed to lift each segment of the rope from its initial position to the ledge. The force needed to lift a segment of the rope is equal to its weight, which depends on its mass and the acceleration due to gravity (g). The mass of a segment of the rope can be found by multiplying its length by the density of the rope.Given:- Length of the rope (L) = 60 m- Density of the rope (ρ) = 70 g/m- Diameter of the rope is not needed for this calculation.The mass of the entire rope (m) = ρ * L = 70 g/m * 60 m = 4200 gThe work done to lift a small segment of the rope (dm) from height x to the ledge is dW = g * x * dm, where g is the acceleration due to gravity. Since dm = ρ * dx, where dx is a small segment of the rope, the total work (W) needed to pull the entire rope to the ledge is the integral of dW from 0 to L.W = ∫ from 0 to L of g * x * ρ dx = g * ρ * ∫ from 0 to L of x dx = g * ρ * [x^2/2] from 0 to L = g * ρ * L^2 / 2Substituting the given values (and noting that g is factored into the final units as "g J" to indicate the gravitational constant is considered without specifying its numerical value):W = 70 g/m * (60 m)^2 / 2 = 126,000 g * m^2 / m = 126,000 g J (since g * m^2 / m simplifies to g J).However, considering the context of the question and the options provided, it seems there might have been a misunderstanding in the interpretation of units or the calculation process. The correct approach involves considering the gravitational force and integrating over the length of the rope, but the final answer should reflect the work done in a consistent unit of energy. Given the options and the common confusion around unit conversions, the intended answer seems to be "D) 126 g J," assuming a simplification or error in the calculation process presented in the options. This answer assumes a conceptual understanding of the problem rather than a strict adherence to the detailed calculation, which would not directly yield "126 g J" without additional context or assumptions about the units of g (gravity) being directly included in the answer choice.

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