# Weight (physics): calculation, units, examples, exercises

We explain weight (physics): calculation, units, examples, exercises. The **weight** is the force with which the Earth attracts objects to its surface. Every time an object is dropped, it falls to the ground, it is not able to climb on its own, nor is it weightless halfway through, which is because the Earth attracts it. weight formula units

All objects invariably attract each other, even the smallest ones, only the magnitude of the force with which they do so is proportional to the mass. This means that objects with small mass exert little force on others, but celestial bodies such as the Earth are capable of exerting a very large force. weight formula units

The Earth keeps the Moon orbiting around it thanks to this attractive force, which is called *gravitational attraction* when it comes to objects that are far from the Earth’s surface, and *weight* when objects are close.

From this it follows that the force of gravity does not require objects to be necessarily in contact with each other in order to act: that is why it is said to be a force of action at a distance.

Objects continue to have weight even if they are at a certain height above the ground and the more massive they are, the greater this weight will be. weight formula units

The great English scientist Isaac Newton was the first to give an explanation about this question, through the universal law of gravitation that bears his name and that since then has served to understand how objects interact with mass. This is very important, since any object on the planet has weight.

**Units of weight**

The International System of SI units has the *newton* as the unit for weight , named after Isaac Newton. This is the unit for measuring forces of all kinds. weight formula units

The newton, abbreviated N, is defined as the force necessary for an object with a mass of 1kg to acquire an acceleration of 1m / s ^{2} . Besides the newton, there are other units of force in common use, for example the following:

**The kilogram-force weight formula units**

The **kilogram-force** or kilopond, abbreviated kg-f or kp, although commonly called kg without more, is the force that the Earth exerts on an object that is at the height of sea level and at 45º north latitude. It is necessary to specify the location, since as it was said, the gravitational field experiences variations with the height and the latitude.

When someone says that he weighs 45 kg, what he really means is that his weight is 45 kg-f, because the kilogram is the unit reserved for mass. weight formula units

The equivalence between kg-f and N is: 1kg-f = 9.8 N

**Pound-force weight formula units**

The **pound-force** , abbreviated lb-f, is also a unit of force that is analogous to kg-f, since it is the force that the Earth exerts on an object of 1 lb mass. And as with the kg-f, there is no problem with the values when you are on Earth, that is, an object of mass 1 lb, weighs 1 lb-f.

The equivalence in lb-f and N is: 1 lb-f ≡ 4.448222 N.

**Weight calculation and formula**

The weight of an object is proportional to its mass. The greater the mass, the greater the weight. weight formula units

The formula to find the magnitude of the weight (W) (or also W, as it is sometimes denoted by *“weight”* in English) is very simple:

W = mg

Where **m** represents the mass of the object and W is the magnitude of the acceleration of gravity (intensity of the gravitational field or gravity), approximately constant and whose value is taken as 9.81 m / s ^{2} for the most frequent calculations.

Weight is a vector and bold letters are used to distinguish between a vector and its magnitude. In this way, when we talk about W we understand that it is the numerical value and when we write W we refer to the vector: weight formula units

**W**= m ∙ **g**

The **g** with a bold letter is the Earth’s gravitational field, that is, the influence that the Earth exerts on the space that surrounds it, regardless of whether or not there is another body that perceives it. Any object with mass has its own gravitational field, be it small or large. weight formula units

The intensity of the Earth’s gravitational field **g** is not entirely constant. It has small variations that arise mainly because the Earth is not a perfect sphere and also due to local height and density differences . But for most applications, the value 9.81 m / s ^{2} works very well.

Other celestial bodies have their own characteristic gravitational field, therefore the acceleration of gravity differs depending on the planet or satellite. The same object would have a different weight in each one, hence weight is not a characteristic property of things, but of matter in general. weight formula units

**Weight as vector weight formula units**

Weight is a vector and therefore has magnitude, direction, and sense. In the vicinity of the earth’s surface, the weight is a vector in the vertical direction and the direction is always downwards.

The vertical direction is usually named as the *y* or *z* axis , and the downward direction is assigned a + or – sign to distinguish it from the upward direction. The choice depends on the location of the origin. In the image below, the origin was chosen at the point from which the apple falls: weight formula units

The unit vector **j** , a vector of magnitude equal to 1, is used to mark and distinguish the vertical direction. In terms of this vector, the weight is written like this:

**W**= mg (- **j** )

Where a negative sign is assigned to the downward direction.

**Differences between weight, mass and volume weight formula units**

These three concepts are often confused, but reviewing the characteristics of weight, it is easy to differentiate it from mass and volume . weight formula units

To begin with, the weight depends on the gravitational field where the object is. For example, on Earth and on the Moon the same thing has a different weight, although the number of atoms that compose it remains constant.

Mass is a scalar quantity, related to the number of atoms that make up the object and is evidenced by the resistance that the object has to change its motion, a property called *inertia* .

For its part, volume is the measure of the space that an object occupies, another scalar quantity. Two objects with the same volume do not weigh the same, for example an iron cube weighs much more than a polystyrene cube of the same dimensions.

In summary: weight formula units

- Mass is related to the amount of matter that a body has.
- Weight is the force exerted by the Earth on this mass, proportional to it.
- Volume is the space occupied by matter.

It should be noted that being scalar quantities, neither mass nor volume have direction or sense, but only numerical value and an adequate unit. On the other hand, the weight, being a vector, must always be expressed correctly indicating the magnitude, the unit, the direction and the sense, as in the previous section.

**Weight Examples**

All objects on Earth have weight, you can even “weigh” objects that are not on Earth, such as other planets or the Sun , although by indirect means, of course.

As the range of weights is very large, scientific notation (in powers of 10) is used to express some that are very large or very small: weight formula units

-The Sun: 1,989 × 10 ^{30} kg-f

– Jupiter : 1,898 × 10 ^{27} kg-f

-A mosquito: 2.0 × 10 ^{-5} N

-Babies: 34.3 N

-A child: 353 N

-Adult person: 65 kg-f

-An adult elephant: 5.5 × 10 ^{3} kg-f

-Blue whale: 1.0 × 10 ^{6} N

**Exercise resolved**

A box of mass 20 kg rests on a table.

a) Find the weight of the box and the normal force that the table exerts on it. weight formula units

b) Another 10 kg box is placed on top of the first one. Find the normal that the table exerts on the 20 kg box and the normal that it exerts on the smaller box.

**Solution to**

It is convenient to make a free-body diagram on the box, which consists of drawing the forces that act on it.

In this situation, the smallest box is not on top yet, therefore there are only two forces: the first is the weight W that is drawn vertically downwards, as indicated in the preceding sections, and then there is the normal **N** , which is the perpendicular force exerted by the table and prevents the box from falling.

Given that the box is in static equilibrium under these circumstances, it is reasonable to conclude that the magnitude of the normal is the same as that of the weight, so that it can compensate, therefore: weight formula units

N = mg = 20.0 kg x 9.8 m / s ^{2} = 196 N; directed vertically upwards.

For its part, the weight is P = 196 N directed vertically downwards.

**Solution b**

Now new free-body diagrams are made on both objects. For the big box things change a bit, since the small box exerts force on it.

The forces are as follows: **N** and **W** are respectively the normal exerted by the table and the weight on the 20.0 kg box, that did not change. And the new force exerted by the small box is **N **_{1} , the normal due to contact with the upper face of the large box.

As for the small box, it receives the normal **N **_{2} , exerted by the upper face of the large box and of course its weight **W**_{2} . Since the boxes are in static equilibrium: weight formula unitsweight formula units

N _{2} – W _{2} = 0

N – N _{1} – W = 0

From the first equation we have that N _{2} = P _{2} = 10 kg x 9.8 m / s ^{2} = 98 N. By law of action and reaction, the magnitude of the force that the small box receives is the same that it exerts on the big box, then:

N _{2} = N _{1} = 98 N

From the second equation, the normal N exerted by the table on the large box is cleared, which in turn has the small box on top:

N = N _{1} + W = 98 N + 196 N = 294 N

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