Simple Conceptual Force Problem

An object of mass m is hanging by a string from the ceiling of an elevator. The
elevator is accelerating upward. What is the tension in the string?
A) less than mg
B) exactly mg
C) greater than mg

I think it would be greater than mg, because in order to accelerate upward, your net force must be higher than mg. Is this correct?

You are correct. C) is the answer.
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It is interesting to consider some other cases as well:

When the elevator moves with constant speed, the tension will be exactly mg. It doesn't matter whether the motion is up or down.

If the elevator is is free fall, the tension is zero.