Question

WorkLifting equipment A rock climber is about to haul up 100 N (about 22.5 lb) of equipment that has been hanging beneath her on 40 m of rope that weighs 0.8 N/m. How much work will it take? (Hint: Solve for the rope and equipment separately, then add.)

Accepted Answer

Identify the two components of the work to be done: the work to lift the rock climber's equipment (100 N) and the work to lift the rope itself, which has weight distributed along its length. Calculate the work done to lift the equipment. Since the equipment weighs 100 N and is lifted a distance of 40 m, the work done on the equipment is given by the formula $$W_{equipment} = F 	imes d$$, where $$F = 100 N $$and $$d = 40 m. $$For the rope, recognize that its weight is distributed uniformly along its length, with a linear weight density of 0.8 N/m. The total weight of the rope is $$0.8 	imes 40 N$$, but since the rope is lifted gradually, the force varies with the length lifted. Set up an integral to calculate the work done lifting the rope. Let $$x$$ represent the length of rope lifted at any moment (from 0 to 40 m). The weight of the rope segment being lifted at position $$x is$$ $$0.8 	imes (40 - x) N$$, and the distance it is lifted is $$x$$ meters. The work done on a small segment $$dx is$$ $$dW = 0.8 	imes (40 - x) 	imes dx. $$Integrate the expression for $$dW$$ from $$x=0 to$$ $$x=40 to $$find the total work done lifting the rope: $$W_{rope} = \$$int_0$$^{40} 0.8 	imes (40 - x) \, dx. $$Finally, add the work done lifting the equipment and the work done lifting the rope to find the total work.

Work

Lifting equipment A rock climber is about to haul up 100 N (about 22.5 lb) of equipment that has been hanging beneath her on 40 m of rope that weighs 0.8 N/m. How much work will it take? (Hint: Solve for the rope and equipment separately, then add.)

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Concetti chiave

Work Done by a Force

Variable Force Due to Rope Weight

Integration to Calculate Work with Variable Force