### Video Transcript

An object is attached to a spring below it and another spring above it, as shown in the diagram. The other ends of the springs, which are not attached to the object, are attached to fixed surfaces. A helium balloon is also attached to the object. The weight of the object is 900 newtons and the balloon provides an upward force of 100 newtons. The object is at rest and applies a downward force of 500 newtons on the spring that is connected to the lower surface. What force does the spring that is connected to the upper surface apply to the object? Consider downward to be the positive direction.

Okay, so, in this question, we’ve got a lot of information that we need to decode from this paragraph. Now firstly, we’ve been told that there is an object that is attached to a spring below it and another spring above it. And it looks like this is the object in our diagram. There is a spring below it and another spring above. Now the springs on either side of the object are also attached to fixed surfaces. So, these two are the fixed surfaces in the diagram. And finally, we also know that there is a helium balloon attached to the object.

Now we’ve been told that the weight of the object is 900 newtons. So, let’s label that on the diagram. The weight of the object will naturally act in a downward direction. So, we can draw an arrow representing this weight. And let’s label the weight as a 900-newton force. Now we’ve also been told that the balloon, the helium balloon attached to the object, provides an upward force of 100 newtons. And, of course, this is the force exerted by the balloon onto the object. And so, we can say that there is an upward force exerted by the helium balloon onto the object with a magnitude of 100 newtons.

Now other information that we’ve been given is that the object is at rest. In other words, the object is not moving. It stays in exactly the same position. Well, if the object is not moving, then we can say that the velocity of the object is zero. We can label that as 𝑣 is equal to zero. Now the reason that this is important is because if the velocity of the object is zero, then that must mean that the acceleration of the object is also zero. Because the velocity of the object is not changing. And so, the object is not accelerating.

Okay, so, we’ve figured out that the object being at rest means that its velocity must be zero. And so, its acceleration must also be zero. But why do we care about this? Well, the reason that we bring up the object’s acceleration is because we can then recall Newton’s second law of motion. Newton’s second law of motion tells us that the net or resultant force on an object is equal to the mass of that object multiplied by its acceleration.

Now whatever the mass of the object may be, which we don’t really care about at the moment, we’ve just realised that the acceleration of the object is zero. And so, if we’ve got the mass of the object multiplied by zero on the right-hand side, then the whole of the right-hand side is zero. And hence, the left-hand side is zero as well. So, using Newton’s second law of motion, we’ve just calculated that the net force on this object is going to be zero. Let’s write that down over here because that will become important to us really soon.

Now as an aside, we could’ve calculated the fact that the net force on the object is zero using Newton’s first law of motion, which tells us that any object at rest will stay at rest and any object in motion will continue to travel at the same constant velocity unless an unbalanced force acts on it. And so, in this case, because we know that the object is at rest, we know that all of the forces acting on the object must be balanced. Which, in other words, tells us that the net force on the object is zero. So, we’ve used a different one of Newton’s laws to come to the same conclusion.

Now we’re making good progress in terms of decoding the information given to us in this paragraph, so let’s continue doing that. We’ve figured out the weight of the object, the force applied by the balloon on the object in the upward direction, and the fact that the net force on the object must be zero. Aside from this, we have been told that the object applies a downward force of 500 newtons on the spring that is connected to the lower surface. So, this is the spring connected to the lower surface and we’ve been told that the object applies a downward force of 500 newtons onto that spring. So, we can label that force.

Now the reason that this is useful information is because based on this force, we can work out the force exerted by the spring onto the object. Because this is the force exerted by the object onto the spring. And the way that we can work this out is using Newton’s third law of motion. Newton’s third law of motion tells us that if an object, let’s say object A, exerts a force on another object, object B, then that second object B will exert an equal and opposite force on the first object A.

So, in this situation, we know that this object, which we’ve called the object confusingly, is exerting a force on another object, which is actually the spring in this case. And that force is a downward force of 500 newtons. Therefore, by Newton’s third law of motion, the spring will exert an equal and opposite force on the object. In other words, the spring will exert an upward force of 500 newtons onto the object. So, the force exerted by the spring is in the opposite direction, that’s upwards, but of the same magnitude 500. And hence, we’ve worked out another one of the forces acting on the object.

So, let’s continue with decoding the information in this paragraph. Let’s look at what this question is actually asking us to find. The question asks us, what force does the spring that is connected to the upper surface apply to the object? In other words, what force does this spring here exert onto our object? And as well as this, we’ve been told to consider downward to be the positive direction. In other words, we can put down a label saying that the downward direction is positive. And therefore, the opposite direction to that, the upward direction, must be negative. So, any forces acting in the downward direction can be represented as positive and any forces acting upward can be represented with a negative sign.

Now let’s label the unknown force that we’re trying to find, which is the force exerted by this upper spring onto our object. Because we don’t yet know the direction in which that force is going to be acting, let’s just say for simplicity that the force exerted by the spring is acting in a downward direction. Now because we’ve cluttered up our diagram, let’s just draw the arrow down here. But in reality, of course, the force would be exerted at this point here where the spring connects with the object.

But anyway, so, the reason that we’ve chosen our blue arrow to be in the downward direction is because the positive direction is the downward direction. And actually, if the spring does end up exerting an upward force and we’ve got the direction of the arrow wrong, then this is not a problem. When we calculate what the magnitude of this force is we’ll find our value to be negative, which will tell us that we’ve got the direction of the arrow the wrong way around.

So, let’s say that this force has a magnitude of 𝑥, which is what we’ll try and find now. And the way that we’re gonna do this is to add up all of the forces acting on the object. We’re gonna ignore all of the forces acting on anything else, such as the two springs or the helium balloon, and only focus on the forces acting on the object. Because this way, when we add up all the forces acting on the object whilst also accounting for the direction in which they act, then the result of the adding up will give us the net force on the object. And as we’ve already seen, the net force on the object must be zero. So, based on this information, let’s start adding everything up.

Let’s start with the force exerted by the balloon onto the object. We know that that force has a magnitude of 100 newtons. But remember, it’s acting in the upward direction. Therefore, we say that this force is negative 100 newtons. And then, to this, we can add the upward force exerted by the lower spring, which, remember, we calculated using Newton’s third law of motion. So, to our negative 100 newtons, we add a negative 500-newton force because, once again, this force is acting in the upward direction.

And then, we can add the weight of the object, which is going to be positive 900 because it’s acting downwards. And finally, we can add the force that we’re trying to find. Since for now we’re saying that it acts in a downward direction possibly potentially, we’re going to add 𝑥 to this sum. So, we’ve accounted for all of the forces acting on the object. And hence, that must be equal to the net force on the object, or the resultant or the overall force on the object. And like we said earlier, this force is going to be zero newtons. And so, we replace 𝐹 subscript net with zero newtons.

Now all we need to do is to rearrange to solve for 𝑥. But before we do, we can add negative 100 newtons to negative 500 newtons to 900 newtons. And all of that together becomes positive 300 newtons. Then, to solve for 𝑥, we subtract 300 newtons from both sides. This way, the positive 300 newtons cancels with the negative 300 newtons on the left, and we’re just left with 𝑥. And on the right-hand side, we’ve got zero newtons minus 300 newtons, which ends up being negative 300 newtons. And so, we found the value of 𝑥, the force that we’re trying to find.

And we can see that because this value is negative, we actually did get the direction of this downward arrow wrong. In other words, then, the force exerted by this spring onto our object is actually not in a downward direction, but rather it is an upward force of 300 newtons. But anyway, because the question is telling us that downward is positive, and, therefore, upward is negative, we can then say that the force exerted by the spring that is connected to the upper surface onto our object is negative 300 newtons.