Weight Definition

Weight formula

The formula for the weight of an object is

$$W=mg,$$

where \( W \) is measured in \( \mathrm N \), \( m \) is the mass of the object measured in \( \mathrm{kg} \) and \( g \) is the gravitational field strength measured in \( \mathrm m/\mathrm s^2 \).

You may have noticed that the units for gravitational field strength \( \mathrm m/\mathrm s^2 \) are the same as the units for acceleration. Gravitational field strength is also known as gravitational acceleration - it is the acceleration of an object due to gravity. Maybe you can now see the similarity between the weight equation and Newton's second law equation, which is,

$$F=ma,$$

where \( F \) is the force required to act on an object of mass \( m \) to give it an acceleration \( a \). They are in fact the same equation, but the weight equation is for the specific situation of when an object feels a force due to a gravitational field.

When we talk about the weight of object's on the earth's surface, we must use the value of \( g \) on the Earth's surface, which is approximately \( 9.8\,\mathrm m/\mathrm s^2 \). As mentioned above, weight depends on the gravitational field that the object is in. On the surface of the moon, the gravitational field strength is approximately \( 6 \) times less than that on the Earth's surface, so the weight of an object on the moon will be \( 6 \) times less than its weight on the Earth.

Difference between mass and weight

The concepts of mass and weight are often confused with each other, but they are very different in the context of physics. The mass of an object is a measure of the amount of matter or the amount of stuff in the object. Mass does not only depend on the amount of matter but also on the density of this matter; objects of the same volume can have different masses. On the other hand, the weight of an object is the force acting on the object due to gravity. The mass of an object is the same everywhere whereas the weight changes depending on the strength of the gravitational field.

There is a directly proportional relationship between weight and mass, as can be seen from the weight formula. The greater an object's mass, the greater its weight will be. The proportionality constant is the gravitational field strength, \( g \). However, we must remember that weight is a vector quantity - it has a magnitude and a direction - whereas mass is simply a scalar quantity and only has a magnitude. The reason that mass is transformed into the vector quantity weight after being multiplied by the gravitational field strength \( g \), is because \( g \) is more than just a simple multiplicative constant, it is also a vector quantity.

At every point in a gravitational field, the gravitational field strength vector points in the direction where a mass will feel a force. For instance, on the Earth, the gravitational field vector always points towards the Earth's centre. However, at nearby points, the \( g \) vectors can be approximated as a parallel because the distance between two points is usually negligible compared to the circumference of the Earth (approximately \( 40,000\,\mathrm{km} \). Even though in reality they point in minutely different directions, for all practical purposes they can be treated as parallel.

Calculation of weight

We can use everything we have learned about weight in many different practice questions.

Examples of weight in science

There are some interesting situations that arise when objects move under the influence of gravity. An example of this is weightlessness, which is the state of apparently not being acted on by gravity. You feel weightless when there is no reaction force against your weight. When we stand on the ground, we feel the ground pushing upwards against our bodies with a force that is equal and opposite to our weight.

Rollercoasters

You may have been on a rollercoaster or a fairground ride that involves a vertical drop and have experienced what is called free fall, which is when you feel weightless while falling. As you fall, the only force acting on you is gravity, but you cannot feel it as there is no reaction force acting in the opposite direction. In fact, this definition of free fall is only used colloquially because while you fall there is actually the force due to air resistance acting upwards on you to oppose your motion. However, this force is relatively small at low speeds and so can be ignored. If you were to jump off the lip of a crater on the moon, you would experience true free fall (until you hit the ground) as there is no atmosphere on the moon.

Fig. 3 - You can experience the sensation of 'free fall' on some rollercoasters.

Astronauts in space

You will have surely seen images of astronauts floating around in space shuttles while orbiting the Earth. The weightlessness felt by astronauts in space is actually identical to the feeling of free-fall on a rollercoaster! The astronauts are falling down towards the Earth, but because their space shuttle moves at such a great speed tangential to the Earth's centre, they effectively keep missing the Earth. The tangential speed (the speed in a direction perpendicular to the direction of Earth's centre) of the astronauts in the shuttle, combined with the curvature of the earth means that as they are pulled towards the earth by gravity, the Earth is actually curving away from them.

Fig. 4 - Astronauts feel weightless when orbiting the Earth in a spacecraft but the Earth still exerts a gravitational force on them