Question(1 point)
A wrecking ball hangs from a 50 - ft cable of density attached to a crane. Calculate the work done if the crane lifts the ball from ground level to 50 ft in the air by drawing in the cable.
ft-lbs.
Studdy Solution
STEP 1
1. The wrecking ball weighs lbs.
2. The cable has a density of lbs/ft.
3. The cable is ft long.
4. Work is calculated as the integral of force over distance.
STEP 2
1. Calculate the work done to lift the wrecking ball.
2. Calculate the work done to lift the cable.
3. Sum the work done for both the wrecking ball and the cable.
STEP 3
Calculate the work done to lift the wrecking ball:
The work done to lift the wrecking ball is given by the formula:
Substitute the given values:
Calculate the work:
STEP 4
Calculate the work done to lift the cable:
Consider a small segment of the cable at a distance from the top. The weight of this segment is:
The work done to lift this segment to the top is:
Integrate over the length of the cable from to ft:
Calculate the integral:
STEP 5
Sum the work done for both the wrecking ball and the cable:
The total work done is:
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