The moon is a weird and wonderful place. Only a few people in the history of our species have ever set foot on it. You may have seen videos of astronauts leaping effortlessly across the Luna landscape, or hitting golf balls huge distances in front of the backdrop of the moon's many craters. All of this is possible because the astronauts weigh much less on the moon than on Earth due to the moon's weaker gravitational pull. However, this is not a trick to lose weight without going on a diet - when astronauts return to Earth they will be the same weight as before! This may seem obvious, but the concepts of weight and mass are easy to confuse. Read on to learn the definition of weight and more about how it is related to mass.
Definition of weight in science
Weight is the force acting on an object due to gravity.
The weight of an object depends on the gravitational field at the point in space where the object is. Weight is a force so it is a vector quantity, which means that it has direction as well as magnitude. It is often convenient to represent the force due to the weight of an object by a free-body diagram.
Weight always acts down from the centre of mass of an object, towards the centre of the Earth. (This will of course be different if you are on a different celestial body, such as Mars or the moon.) A cross-section of a car is shown below, its weight acts directly downwards from its centre of mass.
Fig. 1 - The force due to the weight of a car acts directly downwards from its centre of mass
The centre of mass of an object or system is the point at which all of the mass of the object can be considered to be.
The centre of mass is not always the geometric centre of the object! This discrepancy is usually due to a non-uniform distribution of mass inside an object or system.
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.
It is not entirely correct that the mass of an object is always the same. The rest mass of an object is always constant, but the relativistic mass of an object increases as its speed increases (relative to an observer). However, this effect is often negligible and only becomes relevant when an object moves close to the speed of light. The relativistic mass of any object approaches infinity as an object's speed approaches light speed \(c\) or \(3 \times 10^8\,m/s\), which is why no object with mass can reach or surpass the speed of light!
You will not study objects moving near the speed of light in GCSE but if you are interested you should research the special theory of relativity. This theory also describes the equivalence of mass and energy through physic's most famous equation, \( E=mc^2 \). In particle accelerators, for instance, high-energy particles are smashed into each other in order to create more particles - energy is converted into mass.
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.
Question
A large apple has a weight of \( 0.98\,\mathrm N \) on the surface of the Earth. What is the mass of the apple?
Solution
For this question, we need to use the weight formula, which is
$$W=mg.$$
The question asks for the mass of the apple, so the formula must be rearranged to find mass in terms of weight and gravitational field strength,
$$m=\frac Wg.$$
The weight of the apple is given in the question and the gravitational field strength on the surface of the Earth is \( 9.8\,\mathrm m/\mathrm s^2 \), so the mass of the apple is
$$m=\frac{0.98\,\mathrm N}{9.8\,\mathrm m/\mathrm s^2}=0.1\,\mathrm{kg}.$$
Question 2
A weightlifter attempts to lift a \( 40\,\mathrm{kg} \) dumbbell off the ground. If she exerts an upward force of \( 400\,\mathrm N \) on the dumbbell, will she be able to lift it off the floor?
Solution 2
For the weightlifter to lift the dumbbell off the floor, she needs to exert an upwards force on it that is greater than the downward force due to the weight of the dumbbell. The weight of the dumbbell can be calculated as
$$W=mg=40\,\mathrm{kg}\times9.8\,\mathrm m/\mathrm s^2=392\,\mathrm N.$$
The downward force due to the weight of the dumbbell is \( 392\,\mathrm N \) and the upward pulling force that the weightlifter exerts is \( 400\,\mathrm N \). As \( 400>392 \), the weightlifter will successfully lift the dumbbell!
Question 3
An astronaut has a weight of \( 686\,\mathrm N \) on Earth. What is her weight on the moon? The gravitational field strength on the surface of the moon is \( 1.6\,\mathrm m/\mathrm s^2 \).
Solution 3
Let us first define the following quantities:
- The astronaut's weight on the Earth is \( W_{\mathrm E} \)
- The astronaut's weight on the moon is \( W_{\mathrm M} \)
- The gravitational field strength on the Earth's surface is \( g_{\mathrm E} \)
- The gravitational field strength on the moon's surface is \( g_{\mathrm M} \)
The weight equation for the astronaut on Earth can be written as
$$W_{\mathrm E} =mg_{\mathrm E},$$
so the astronaut's mass is
$$m=\frac{W_{\mathrm E}}{g_{\mathrm E}}.$$
Now, for the astronaut on the moon, the weight equation is
$$W_{\mathrm M}=mg_{\mathrm M},$$
and her mass is
$$m=\frac{W_{\mathrm M}}{g_{\mathrm M}}.$$
The mass of an object is always the same so we can equate the two expressions to get
$$\frac{W_{\mathrm E}}{g_{\mathrm E}}=\frac{W_{\mathrm M}}{g_{\mathrm M}},$$
which can be rearranged to give the weight of the astronaut on the moon as
$$W_{\mathrm M}=\frac{W_{\mathrm E}g_{\mathrm M}}{g_{\mathrm E}}=\frac{686\,\mathrm N\times1.6\,\mathrm m/\mathrm s^2}{9.8\;\mathrm m/\mathrm s^2}=112\;\mathrm N.$$
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.
An orbit is the curved path of a space shuttle or celestial object around a star, planet, or moon. It is the tangential velocity of any orbiting object that stops them from simply being pulled down with any celestial body and colliding with it!
Fig. 4 - Astronauts feel weightless when orbiting the Earth in a spacecraft but the Earth still exerts a gravitational force on them
Weight Definition - Key takeaways
- Weight is the force acting on an object due to gravity.
- The centre of mass of an object is the point at which all of the mass of the object can be considered to be.
- The mass of an object is a measure of the amount of matter making up the object.
- Weight is a vector quantity.
- Mass is a scalar quantity.
- The weight of an object depends on its position in a gravitational field whereas its mass is the same everywhere.
- The formula for the weight of an object is \( W=mg \).
- There is a directly proportional relationship between an object's mass and its weight.
References
- Fig. 1 - Car free-body diagram, StudySmarter Originals
- Fig. 3 - you experience the sensation of 'free fall' on some rollercoasters (https://upload.wikimedia.org/wikipedia/commons/9/9c/Rollercoaster_expedition_geforce_holiday_park_germany.jpg) by Boris23, Public domain, via Wikimedia Commons
- Fig. 4 - astronauts feel weightless when orbiting the Earth in a spacecraft but the Earth still exerts a gravitational force on them (https://commons.wikimedia.org/wiki/File:STS083-302-036_-_STS-083_-_Candid_views_of_Pilot_Still_floating_in_Spacelab_module_-_DPLA_-_bfaeb0e0e302e29af46e5b7e4d55904c.jpg) National Archives at College Park - Still Pictures, Public domain, via Wikimedia Commons
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