Fill in the blanks. An object that is initially at rest and has two forces acting on it will remain at rest if the forces acting on it have blank and act in blank.
Okay, so in this question, we’re discussing an object which we’ve been told is initially at rest. That means it’s not moving. And as well as this, we’ve been told that it has two forces acting on it. Now, these two forces could be acting in the same direction or in opposite directions. Or they could be acting in any direction really. And as well as this, we know nothing about the magnitudes or sizes of these two forces yet. For all we know, one of the forces could be much larger than the other.
However, what this question is telling us is that this object is initially at rest. So it’s not moving. And it has two forces acting on it. But it still remains at rest. In other words, even though there are two forces acting on it, the object does not move. So in which situation can an object have forces acting on it and yet still not be moving? Well, to help us answer this question, we can recall Newton’s first law of motion. This law tells us that an object at rest remains at rest. And an object moving at a constant velocity continues to move at that same constant velocity unless a net force acts on it. And the reason that this is relevant to our object here is because that object was initially at rest. And it remains at rest.
Now, Newton’s first law tells us that if an object remains at rest, then this must mean that there’s no net force acting on it. Because if there is a net force acting on the object, then the object will change velocity. It will accelerate. And so what we can gather from this is that the net force on our object has to be zero. The object that started at rest remains at rest. And therefore, there must be no net force acting on it.
Now, remember that Newton’s first law is only referring to the net or resultant force on the object. And actually, we know that there are two forces acting on our object at the same time. So how is it that two forces acting on an object can give a resultant or net force of zero?
Well, let’s recall that the net force on an object is simply a combination of all other forces acting on the object. In other words, if we call the two forces acting on the object 𝐹 one and 𝐹 two, then we can say that 𝐹 one plus 𝐹 two, which is how we combine the two forces acting on the object, is equal to the net force on the object, which is zero in this case. And hence, we can take this equation that we’ve just found and rearrange it.
And one way to do this is to subtract 𝐹 two from both sides of the equation from the left and from the right. Which means that on the left-hand side, we’ve got a negative 𝐹 two plus 𝐹 two. These add together to give zero, leaving us with just 𝐹 one on the left-hand side. And on the right-hand side, we have zero minus 𝐹 two, which just ends up being negative 𝐹 two. So what we find then is that 𝐹 one is equal to negative 𝐹 two.
In other words, firstly, 𝐹 one and 𝐹 two act in opposite directions. And we know this because the positive force 𝐹 one, which we can say is acting in this direction towards the right, is equal to negative 𝐹 two. Now, the negative sign tells us that the force 𝐹 two must be acting in the opposite direction to the force 𝐹 one. And so, now we can say that if force 𝐹 one is acting towards the right on our object, which it doesn’t have to of course, we’ve just arbitrarily chosen it to act towards the right. Then 𝐹 two must act in the opposite direction to the left.
And secondly, we can see that the magnitudes or sizes of 𝐹 one and 𝐹 two must be exactly the same because we’re not seeing any multiplication factor on either side of this equation, aside from the negative sign of course. And we’ve already seen that the negative sign just tells us which direction 𝐹 two is acting in. But if 𝐹 one and 𝐹 two didn’t have the same magnitudes, then we would see something like 𝐹 one is equal to negative two 𝐹 two say if force 𝐹 two had twice the magnitude of 𝐹 one or something along those lines. But what we do see is that the magnitudes of 𝐹 one and 𝐹 two are exactly the same.
And so at this point, we figured out that the two forces acting on the object have the same magnitudes and act in opposite directions. Now, this makes sense. Remember, we’re trying to find the two forces acting on our object such that the net force on the object is zero. And the only way that the resultant force on an object can be zero, if there are two forces acting on it, is if the two forces have the same magnitudes but act in opposite directions. So they cancel each other out exactly.
And so what we can say to fill in our blanks now is the following. An object that is initially at rest and has two forces acting on it will remain at rest if the forces acting on it have the same magnitude or size, because these forces need to exactly cancel each other out. And these forces need to act in opposite directions if they are to cancel each other out. So the full statement now reads, an object that is initially at rest and has two forces acting on it will remain at rest if the forces acting on it have the same magnitude and act in opposite directions.