# Video: The Relationship between Mass and Weight

In this video, we will learn how to apply Newton’s second law of motion to define gravitational field strength as the force acting on an object per kilogram of its mass.

10:25

### Video Transcript

In this video, we will be looking at the subtle difference between mass and weight. In everyday language, we often use them interchangeably. They are certainly very closely linked, but they are not the same thing. So let’s take a look at the definitions of mass and weight and understand the difference between them.

Firstly, mass is a measure of the amount of matter or stuff that makes up an object. Now we can see that matter is a technical term and it refers to stuff in the universe, stuff such as atoms or protons and neutrons and electrons which happen to be the building blocks of things with mass. So mass is a measure of the amount of stuff that makes up an object. As well as this, the mass of an object is also a measure of the object’s resistance to acceleration when a certain net force is applied to it.

In other words, let’s think about a block that we have. Let’s say it’s a wooden block with a mass 𝑚. And on this wooden block, we exert a net force of magnitude 𝐹 onto it. Well, in that situation, the block is going to accelerate in the direction of the net force. So we’ll say it has an acceleration 𝑎.

And the relationship between the net force 𝐹, the mass of the block 𝑚, and the acceleration 𝑎 is given by Newton’s second law of motion, which tells us that the net force is equal to the mass of the object multiplied by the acceleration it experiences.

So if we rearrange this equation, we can see that for any particular net force exerted on an object, the acceleration that the object experiences is inversely proportional to the mass. In other words, if the same forces exerted on two different objects with two different masses, the object with larger mass is going to result in a smaller acceleration. And hence, we can think of the object’s mass as its resistance to acceleration when a particular net force is exerted on it. As well as this, mass is measured in kilograms. This is its base unit. So this is what we mean when we talk about mass.

Let’s now take a look at weight. Now weight on the other hand is the force experienced by an object when it’s placed in a gravitational field. In other words, we can think of weight as the gravitational force experienced by an object.

So if we come back to our wooden block with mass 𝑚 and now also consider the fact that it’s in Earth’s gravitational field, so let’s say here is the surface of the Earth, well in that situation the Earth’s gravitational field is going to exert a gravitational force onto the block with mass 𝑚. And this force is going to be an attractive force and it’s going to be acting towards the centre of the Earth. We can label this force as 𝑊 because it’s the weight of the wooden block.

It’s important to realize by the way that if we had considered the Earth in the first situation, then we would also have to consider the weight of the block. But we didn’t think about the block being in a gravitational field. So we could have basically just assumed that in the first case it was floating around in outer space or something like that. However, when an object is placed in a gravitational field, it experiences a gravitational force and that force is the weight of the object.

Now this gravitational force can be calculated by multiplying the mass of the object — in this case our wooden block — by what’s known as the gravitational field strength at the position of the wooden block in this case. Now because we’re considering the wooden block being placed in the gravitational field of the Earth, then in this particular situation 𝑔 is referring to the gravitational field strength of the Earth. It doesn’t necessarily have to be the gravitational field strength of the Earth though. It’s just a gravitational field strength of whatever gravitational field the object is in at that point in time.

And by the way, it’s worth noting that an object doesn’t necessarily only have weight when it’s touching the surface of the Earth for example. It will still experiencing a gravitational force even if it is slightly raised above the surface. And it’s at this point that we can realize that because weight is a gravitational force, it is measured in newtons. In other words, newtons is the base unit of weight.

Now, we’ve seen what happens when there is a net force acting on an object. The object accelerates in the direction of the force. So we can say that if we had this wooden block and the only force acting on it was the weight of the block, then it would accelerate downwards towards the surface of the Earth with an acceleration 𝑎 that we could find by using this equation here, Newton’s second law of motion.

We could say that the net force on the block which in this case was equal to the weight of the block was equal to the mass of the block multiplied by the acceleration experienced 𝑎. But hang on a minute! Haven’t we just said that the weight of the object is equal to the mass multiplied by the gravitational field strength? So this is where we can spot something really interesting: the gravitational field strength of the gravitational field that the object is in is the same thing as the acceleration experienced by the object if the only force acting on the object was the weight of the object.

In other words, gravitational field strength is equivalent to an acceleration. And this is why the quantity 𝑔 is also often known as the acceleration due to gravity. And so, an important thing to realize is that this equation 𝑊 is equal to 𝑚𝑔 is just a special case of 𝐹 is equal to 𝑚𝑎. This is because Newton’s second law of motion refers to any net force acting on an object. And it shows the relationship between the mass of that object and the acceleration experienced by the object. So if we say that the net force on the object is just 𝐹, then it would accelerate with 𝑎 in the same direction as the net force. And this is the relationship that we need to use.

However, this equation 𝑊 is equal to 𝑚𝑔 specifically refers to how the gravitational force 𝑊 — the weight of the object — is related to its mass and the gravitational field strength or acceleration due to gravity.

Now coming back to a distinction between mass and weight, since we said earlier that mass is a measure of the amount of matter or stuff that makes up an object, then this means that we could take the exact same object — let’s say this wooden block with mass 𝑚 — into different gravitational fields. And if this object is still exactly the same object as before that there’s nothing has broken off from it for example, then the mass of that object will still be the same regardless of which gravitational field the object is in.

This is because mass is an intrinsic measure of the amount of stuff that makes up an object. And if we take that same object to different locations, the amount of stuff that makes it up is not going to change. Hence, the mass is not going to change.

However, the weight of an object is very dependent on the gravitational field that the object is placed in. This is because the weight of an object is equal to the mass of that object which stays constant multiplied by the gravitational field strength which can change. In other words, Earth’s gravitational field strength is different to the gravitational field strength of the moon for example. And so, if we took the same object with the same mass from Earth to the moon, it would have a different weight on the moon. And that is one very important distinction between mass and weight. So having seen all of this, let’s take a look at an example question.

An astronaut on Earth, where the gravitational field strength is 9.8 newtons per kilogram, has a mass of 65 kilograms and a weight of 637 newtons. The astronaut is sent to a space station, where the gravitational field strength is 9.5 newtons per kilogram. What is the astronaut’s mass on the space station? What is the astronaut’s weight on the space station?

Okay, so in this question, we’ve initially got an astronaut that’s on the surface of the Earth. And then later, that astronaut is sent to a space station. Now we’ve been told that on Earth where the gravitational field strength is 9.8 newtons per kilogram. So we can say that the gravitational field strength 𝑔 of the Earth subscript 𝐸 is 9.8 newtons per kilogram.

We’ve been told that on Earth, the astronaut has a mass which we’ll call 𝑚 of 65 kilograms and a weight which will be a downward acting force. And we’ll call this 𝑊 and it happens to be 637 newtons. Now based on this information, we need to work out what happens to the astronaut when they are sent to a space station. And we’ve been told that on this space station, the gravitational field strength is 9.5 newtons per kilogram. So we can say that 𝑔 sub 𝑠, which is what we’ll call the gravitational field strength on the space station, is 9.5 newtons per kilogram.

Now we’ve been asked to state what the astronaut’s mass is on the space station and what their weight is on the space station. To do this, let’s recall a relationship between weight, mass, and gravitational field strength. We can recall that the weight of an object 𝑊 is given by multiplying the mass of that object by the gravitational field strength which is also known as the acceleration due to gravity caused by the gravitational field that the object is in.

As well as this, we can recall that mass is a measure of the amount of matter or stuff that makes up an object. Therefore, if we take the same object and put it in a new gravitational field, the mass of that object is not going to change because the object is still made up of the same amount of stuff. Therefore, if we’ve been told that the mass of the astronaut is 65 kilograms on Earth, then the mass of the astronaut is 65 kilograms everywhere, regardless of whether they’re on Earth or in a space station or in some part of outer space.

If the astronaut is made up of the same amount of stuff as earlier, then the mass is going to be exactly the same always. So when we’re asked what the astronaut’s mass is on the space station, we can say that their mass is still 65 kilograms. However, looking at this equation, we can see that the weight of the astronaut will change depending on the strength of the gravitational field that the astronaut is in. And this does change between the Earth and the space station. We can see that the values of 𝑔 are different.

So before we find the weight of the astronaut on the space station, let’s first confirm that this equation does make sense based on the values we’ve been given in the question when the astronaut was on Earth. We can, therefore, say that the weight of the astronaut on Earth, we’ll add this subscript 𝐸 now since we’ve realized that the weight changes based on where the astronaut is. We can say that the astronaut’s weight 𝑊 subscript 𝐸 on Earth is equal to the mass multiplied by the gravitational field strength on Earth 𝑔 subscript 𝐸.

And substituting in values, we see that 637 newtons is equal to 65 kilograms multiplied by 9.8 newtons per kilogram. And the right-hand side of the equation does end up being 637 newtons. Therefore, this equation does work. And we’ve just confirmed this based on the numbers we’ve been given in the question. So now we can move on to applying this to the space station.

We can say that the weight of the astronaut on the space station now — which we’ll call 𝑊 subscript 𝑠, so that’s the downward force when the astronaut is on the space station — is equal to the mass of the astronaut which is still the same multiplied by the gravitational field strength on the space station 𝑔 subscript 𝑠.

Then, we can plug in the values on the right-hand side. The mass is still 65 kilograms. But this time, the gravitational field strength is 9.5 newtons per kilogram. And when we evaluate the right-hand side, we find that the new weight of the astronaut is 617.5 newtons. Therefore, our final answer to this part of the question is that the astronaut’s weight on the space station is 617.5 newtons.

Okay, so now that we’ve had a look at an example question, let’s summarize what we’ve talked about in this video. Firstly, we saw that weight and mass are not the same thing. But they are closely related by the equation 𝑊 is equal to 𝑚𝑔. Where 𝑊 is the weight of an object, 𝑚 is the mass of that object and 𝑔 is the gravitational field strength.

Secondly, we saw that the equation 𝑊 is equal to 𝑚𝑔 is a special case of Newton’s second law of motion 𝐹 is equal to 𝑚𝑎 because Newton’s second law of motion generally refers to the net force on an object, whereas 𝑊 is equal to 𝑚𝑔 specifically refers to the gravitational force exerted on an object or in other words the weight of that object. And this means that 𝑔 the gravitational field strength is also known as the acceleration due to gravity.

And finally, we saw that an object will have the same mass everywhere, assuming of course the object doesn’t break into pieces or anything like that. But that same object may have a different weight in different gravitational fields. This depends specifically on the value of 𝑔, the gravitational field strength. So this is our review on the relationship between mass and weight.