Person in an accelerating elevator.When gravity is present it is indicated by a downward arrow, marked g. When the elevator is accelerating it is indicated by an arrow marked a.

take g as a positive number.) We now want to add up all of the forces on the person, and equate them to ma, where m is the person's mass and a is the person's acceleration. The forces are the person's weight, —mg, and the upward force of the scale on the person's feet, FS. The acceleration is zero, so mg + FS = 0

Solving for FS gives us

By Newton's third law, the force the scale exerts on the person has the same magnitude as the force the person exerts on the scale. Therefore, FS also gives the reading of the scale. In this case it is simply the weight of the person - the expected result.

We now look at the case of no gravity, but with an upward acceleration a. The only force on the person is FS. Applying F = ma gives

If we arrange for the acceleration so its value is equal to g, we have

This is the same result we had in the first case. As far as the person in the elevator is concerned, there is no way to tell the difference between a gravitational field with an acceleration g downward and an upward acceleration g of the reference frame.

To illustrate the point farther, we look at a third case, in which there is gravity, but the elevator is in free-fall. The forces on the person are FS upward and mg downward, and the acceleration is mg downward. This gives us

This tells us that FS is zero. The person is "weightless". The acceleration of the elevator has exactly canceled the gravitational field. For the person inside the elevator, there is no way to distinguish this situation from that of a non-accelerating elevator and no gravitational field. This is the same weightlessness felt by astronauts in orbiting space vehicles (Fig. 8.5). Orbiting objects are also in free-fall, but the horizontal component of their velocity is so great that they never get closer to the ground; they just follow the curvature of the Earth.

If you look carefully at the above discussion, you will see that we have really used the concept of mass in two different ways. In one case we said that a body of mass m, subjected to a force F, will have an acceleration F/m. In this sense, mass is the ability of an object to resist the effects of an applied force. We call this resistance inertia. When we use mass in this sense, we refer to it as inertial mass. The second use of mass is as a measure of the ability of an object to exert and feel a gravitational force. In this context, we speak of gravitational mass. In the same sense, we use electric charge as a measure of an object to exert and feel electrical forces. (So, we should think of gravitational mass as being like a gravitational charge.)

M Pseudo-force in an accelerating space station. In this case the station is accelerating towards Earth (like the free-falling elevator) so the astronaut appears weightless. [NASA]

The principle of equivalence is really a statement that inertial and gravitational masses are the same for any object. If the two masses are equal then they do cancel in the above examples, as we have done. This also explains why all objects have the same acceleration in a gravitational field, a point first realized by Galileo. It is not obvious on the surface of the Earth, since air resistance affects how objects fall. However, a hammer and a feather fall with the same acceleration on the surface of the Moon, where there is no air resistance.

It is important to remember that just because we call both quantities "mass" there is no obvious reason for gravitational and inertial mass to have the same numerical value. In the same way, we expect no equality between the electric charge of an object and its inertial mass. If inertial and gravitational mass are the same, this tells us that gravity must somehow be special. As we will see in the next section, considerable effort has gone into verifying the principle of equivalence.

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