Weight

Weight is the force of gravity acting on an object. The unit of measurement for weight is that of force, which in the International System of Units (SI) is the newton [N].

In many real-world situations, the act of weighing may produce a result that differs from the ideal value provided by the definition used. This is usually referred to as the apparent weight of the object. A common example of this is the effect of buoyancy when an object is immersed in a fluid the displacement of the fluid will cause an upward force on the object, making it appear lighter when weighed on a scale.

The main characteristic of weight is that it is directly proportional to mass and varies only when mass varies, because gravity acceleration is constant. Consequently the Earth attracts every body exerting a different force because different are the masses of the bodies on which it exerts it, but in such a way that the acceleration impressed to each body has always the same value. Moreover, gravity force is always directed vertically downwards. Imagining to look at the Earth from the space we will say that the weight is directed towards the center of the planet and, since the force lies on the direction given by the radius, it will always be perpendicular to the surface in every point.

To be precise, the constant g is not really a physical constant. Actually its value changes according to the altitude and more precisely to the distance of the body from the center of the Earth. In the lessons dedicated to universal gravitation we will see in particular that acceleration of gravity decreases with increasing distance from the center of the Earth, therefore a climber on the top of Everest is subject to a lower acceleration and therefore also a lower weight than the one he perceives when he is at sea level. In any case, weight is greater at the poles while it is less at the equator, and this is due to the fact that the Earth is not perfectly spherical: being more swollen at the equator, on this parallel the distance from the center is greater and therefore the weight is less with the same mass.

But the question is: how much does the acceleration of gravity change, and therefore how much does the weight change, with the same mass and varying the distance from the center of the Earth? In fact we are dealing with percentage variations less than 1% so that g can be treated in good approximation as a constant, provided however to remain close to Earth surface.

Specific weight

The weight of a unit volume of a substance is called specific weight (\(\gamma\) also known as the unit weight) and is equal to:

\[\gamma =\rho g=\left[\dfrac{\textrm{N}}{\textrm{m}^3}\right]\]

where \(\rho\) is the density and \(g\) is the gravitational acceleration.

Commonly the term “specific gravity” is improperly used as a synonym for density and for this reason it is very often referred to as g/cm3 or kg/L or kg/dm3. In this case grams should be understood according to an obsolete definition of grams weight, not grams mass, where 1 gram weight is the weight of 1 gram mass under standard gravity acceleration.

The difference is subtle and for the truth in practice it can often be ignored, but it is important to keep in mind that while density is a ratio between a mass and a volume, the specific gravity is a ratio between a weight (therefore a force) and a volume. Since the weight is equal to the mass multiplied by the acceleration of gravity expressed in g, the specific gravity (expressed in kgweight/m3) and the density have the same value only if we are in a point where the acceleration of gravity is exactly equal to gn (standard gravity that for convention is equal to 9,80665 m/s2 that is 1 g).

Apparent weight

Apparent weight is a property of objects that correspond to how heavy an object is. The apparent weight of an object will differ from the weight of an object whenever the force of gravity acting on the object is not balanced by an equal but opposite normal force. By definition, the weight of an object is equal to the magnitude of the force of gravity acting on it.

This means that even a “weightless“ astronaut in low Earth orbit, with an apparent weight of zero, has almost the same weight as he would have while standing on the ground; this is due to the force of gravity in low Earth orbit and on the ground being almost the same.

An object that rests on the ground is subject to a normal force exerted by the ground. The normal force acts only on the boundary of the object that is in contact with the ground. This force is transferred into the body; the force of gravity on every part of the body is balanced by stress forces acting on that part. A “weightless“ astronaut feels weightless due to the absence of these stress forces. By defining the apparent weight of an object in terms of normal forces, one can capture this effect of the stress forces. A common definition is “the force the body exerts on whatever it rests on.“

The apparent weight can also differ from weight when an object is “partially or completely immersed in a fluid“, where there is an “upthrust“ from the liquid that is working against the force of gravity. Another example is the weight of an object or person riding in an elevator. When the elevator begins rising, the object begins exerting a force in the downward direction. If a scale was used, it would be seen that the weight of the object is becoming heavier because of the downward force, changing the apparent weight.

The role of apparent weight is also important in fluidization, when dealing with a number of particles, as it is the amount of force that the “upward drag force“ needs to overcome in order for the particles to rise and for fluidization to occur.

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