Work done pulling a rope to the top of a building
A 50 m rope with a mass density of 0.2 kg/m hangs over the edge of a 30 meter building. How much work is done pulling the rope to the top? We first visualize the scenario with a sketch. The challenge arises from the fact that different segments of the rope are displaced differently - the segment on the ground requires lifting the full height of the building, while the segment already hanging experiences less displacement.
We approach this problem by dividing the rope into two parts: the part hanging over the edge and the part lying on the ground. For the hanging part, we calculate the force as a function of its mass, which is determined by its density and length, and the gravitational acceleration. This force cts over a distance determined by the segment's vertical position, leading us to calculate the work done through integration over the displacement from the top to the bottom of the building.
For the segment on the ground, since it's uniformly lifted 30 m, the calculation is more straightforward. We multiply the length of this segment by its mass (derived from its density), the acceleration due to gravity, and the height it's lifted.
By adding the work done for both parts, we conclude that a total of 2058 Joules of work is needed to pull the entire rope to the top of the building. This example illustrates the application of basic physics principles to a practical problem, emphasizing the importance of understanding variable forces and displacements in calculating work.
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