Apparent weight

Apparent weight

An object's weight, henceforth called "actual weight", is the downward force exerted upon it by a gravitational field. By contrast, an object's apparent weight is the (usually upward) force (the "normal force", or "reaction force"), typically transmitted through the ground, that opposes the (usually downward) acceleration that a supported object would otherwise experience, preventing it from falling.

It is apparent weight, not the actual weight, that a weighing scale measures. Apparent weight is also responsible for our sensation of the weight of our own bodies. A greater apparent weight results in a heavier or greater sensation of our weight, and vice-versa. An object's apparent weight is equal to its actual weight, unless:

*The object has an acceleration, as in a lift, a rocket, or a rollercoaster.
*Some force other than the earth's gravity and the normal force is acting on the object. This may, for example, be buoyancy, centrifugal force due to the Earth's rotation, magnetic force, or the gravitational force of another astronomical body.

Objects at rest

Suppose that Alice has a mass of 65 kilograms and is standing stationary on the floor. Gravity is pulling her downward with a force of:


We already know from the earlier example that "F"gravity = 637 N. Therefore

:637 N + "F"normal = 195 N


:"F"normal = −442 N

Thus the normal force is 442 N (upwards), so Alice's apparent weight is 442 N, and it feels to Alice as if her weight has decreased by about 30% — even though her "actual" weight (the force exerted on her by gravity) remains unchanged. If Alice were standing on a weighing scale then the weight registered would also be 442 N. This is because a scale does not measure an object's actual weight, but rather measures the force that it exerts on the scale.

The workings above consolidate into the neat formula

:"F"normal = "m"("a" − "g")

We can now more easily examine what happens when "a" takes a range of different values. The formula shows that when "a" = 0 we have "F"normal = −"mg", which is, as expected, just the formula for Alice's actual weight (with the negative sign reflecting the fact that the normal force is upwards).

For increasingly positive "a" (increasing downward acceleration), the absolute magnitude of "F"normal (the magnitude ignoring the sign) steadily decreases, meaning that Alice's apparent weight steadily decreases. Conversely, for increasingly negative "a" (increasing upward acceleration), Alice's apparent weight steadily "increases".

It is important to understand that it is "acceleration", not velocity, that causes changes in apparent weight. In a lift travelling upwards or downards at any "constant" speed – however great – Alice's apparent weight will be the same as if the lift were at rest.


Apparent weight decreases with increasing downward acceleration until eventually "a" reaches "g" (the acceleration due to gravity, 9.81 m/s2), when the formula shows that Alice's apparent weight is "zero". At this point the floor of the lift no longer provides any supporting force at all, the only force acting on her is gravity, and Alice is in free-fall. During free-fall Alice experiences weightlessness.

Free-fall also occurs, of course, if Alice is falling freely through the air (in the absence of any containing lift). Ignoring air resistance there is again no supporting force, and no sensation of weight. Objects in orbit are also in free-fall; the mechanics are explained at Orbit.

"Beyond" free-fall

If "a" increases beyond "g" then the formula shows that "F"normal becomes "positive", meaning that the normal force acts "downwards" and Alice experiences a "negative apparent weight". In fact, what happens is that Alice "falls" to the ceiling of the lift, which she now experiences as a floor, and which now provides a "supporting" force.

Objects accelerating in arbitrary direction

In general, an object's apparent weight is its mass multiplied by the vector difference between the gravitational acceleration and the acceleration of the object. This definition means that apparent weight is a vector that can act in any direction, not just vertically. For example, in a racing car accelerating horizontally at 1 "g", apparent weight acts at an angle of 45 degrees downwards. For the calculation of apparent weight in cases where the two accelerations do not fall on a line it is often essential to utilize vector notation.

The apparent weight for an object under the influence of two arbitrary forces mathbf{F}_{1}, and mathbf{F}_{2}, in the most general case can be calculated by the addition of vectors:



Apparent weight is lessened by buoyancy, which occurs when an object is immersed in a fluid (a liquid or a gas). For example, an object immersed in water weighs less, according to a spring balance, than the same object in air. The apparent weight of a floating object is zero.

This effect is quite different from the accelerating lift examples. A floating or immersed object is not accelerating upwards or downwards, so there can be no net force. In fact, buoyancy provides a supporting force exactly as the ground does. Because this force is diffuse and dispersed over the surface of the body, a feeling of "pseudo-weightlessness" arises when one is floating.

Objects also experience some buoyancy in air, so even in air the normal force (apparent weight) is slightly less than the true force of gravity. The density of air at sea level is about 1.2 kg/m3. Most objects are much denser than air and so the difference is usually small. For an object with the same density as water – about 1000 kg/m3 – the relative effect is about 0.12%. However, for objects of very low density the relative effect can be large. In fact, an object that is lighter than air, such as a helium balloon, has a "negative" apparent weight (as does an object lighter than water when it is forcibly pulled below the surface).

Variations in air pressure cause variations in air density and hence variations in apparent weight. Over a typical range of sea-level pressures this may amount to about a 0.01% change in apparent weight for an object of the same density as water (less for denser objects, more for less dense objects). Extremely accurate measurements must take this into account. [ [ "Buoyancy Correction and Air Density Measurement"] , National Physical Laboratory]

Other factors

In general, any force (other than the normal force) that opposes or augments the downwards force of gravity will have an effect on apparent weight. Some examples are:

*Centrifugal force. Because the earth is a rotating reference frame, objects on the earth's surface experience a small centrifugal force, which increases at lower latitudes (nearer to the equator). This offsets the force of gravity, resulting in a small decrease in the net downward force, and a corresponding small decrease in the balancing normal force. Note that this is a pseudo centrifugal force that is used here because it is convenient to view objects in a reference frame rotating around the Earth, rather than considering the constantly changing directions of the various forces involved. It is intentionally not referred to as centripetal force.

*The gravitational influence of the Sun and the Moon. This varies slightly across the Earth because different locations lie at slightly different distances from these bodies. This alters the net gravitational force on objects at the Earth's surface by a small amount, depending on their location on the Earth and the relative positions of the Sun, Earth and Moon. The effect of other astronomical bodies is vanishingly small.

*Magnetism. Strong magnetic fields have even been used to levitate frogs! []


See also

* Force
* Fundamental force
* Gravitational force
* Weight

External references

* [ Physical demonstration with applet]
* [ Exploratorium explanation ]
* [ Elevator problem]

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