# Special case of work done by two opposing forces of equal magnitude on a body possesing inertial motion

Consider an elevator moving upward with constant velocity. The constant velocity is maintained by an upward tensional force equal to weight. If we add the total work done on the elevator it will be zero since work done by two equal and opposite forces cancel out. But the elevator's P.E is increasing. How and who does the work? I think no change P.E takes place i.e. only when we have removed the upward force we can have PE stored in the system. Is this true? Also, can we say that the upward force is *doing any work * because although the object is displacing in the direction of force but it is not causing it.

A common misconception. The elevator does negative work on the motor (the force of the elevator on the rope is in the opposite direction to the velocity), while the motor does positive work on the elevator (force and velocity in the same direction).

So energy is transferred from the motor to the elevator, where it becomes potential energy (in the earth-elevator system: the force of gravity on the elevator is in the opposite direction to the velocity of the elevator relative to the earth...)

The total work done on an object by all forces is equal to the change in its kinetic energy.

If you take the sum of all the forces on an object and multiply it by the distance traveled, you will have calculated the change in kinetic energy, not potential energy.

$(F_1 + F_2 + \cdots + F_n)\cdot\Delta x = \Delta K$

For your elevator, the forces sum to zero, so the change in kinetic energy is zero ($\Delta K = 0$). Thus, the elevator moves with a constant velocity.

We can group these forces into conservative and non-conservative forces. Conservative forces give rise to potential energies (gravity, electric fields, among others). The motor does not produce a conservative force.

$(F_{conservative} + F_{non-conservative})\cdot\Delta x = \Delta K$

In our case, the conservative force is gravity, and the non-conservative force is the motor.

$(F_{gravity})\cdot\Delta x + (F_{motor})\cdot\Delta x = \Delta K$

The work done by a conservative force results in changes in potential energy.

$-\Delta P + (F_{motor})\cdot\Delta x = \Delta K$

The negative sign comes from the fact that if gravity (for example) does positive work on an object by pulling it down to the ground, the potential energy of that object decreases since it goes to a lower elevation.

$(F_{motor})\cdot\Delta x = \Delta K + \Delta P$

So, the motor doing work can change both the potential and kinetic energy of the elevator. In your question, the elevator has a constant speed, so $\Delta K = 0$. Thus,

$(F_{motor})\cdot\Delta x = \Delta P$

The motor's work does indeed change the potential energy of the elevator.