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The extra tension depends not on the velocity of the elevator, but on its acceleration.

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Q: How the tension can be greater than gravity yet the elevator is moving downwards?
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How do you draw an elevator free body diagram?

Draw an arrow pointing upwards for the tension force and an arrow pointing downwards for the weight of the elevator which will be its mass times gravity (mg). Also, draw another arrow pointing downwards for any mass that may be inside the elevator (another mass times gravity arrow but for a separate weight) and add that value to that of the weight of the elevator. Depending on the direction that the elevator is moving (up or down) draw another arrow respectively and label it "a" for acceleration.


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