An object with a weight 𝑊 is equal to 20 newtons is attached to a string. The other end of the string is attached to a spring, as shown in the diagram. The spring is stretched until it comes to rest. How much vertically downward force does the string apply to the spring? How much vertically upward force does the spring apply to the string? How much vertically upward force does the string apply to the object?
Okay, so, in this question, we’ve got three components all attached together. Firstly, we’ve got the spring. Secondly, we’ve got the string. And thirdly, we’ve got the object. Now we’ve been told that the object has a weight 𝑊 which happens to be 20 newtons. And as well as this, we know that the object is attached to the string which is then attached to the spring. And the spring is extended until it comes to rest. But then if the spring is at rest, then that means that the string and the object, which are attached to the spring, must also be at rest. Therefore, this whole system is at rest.
Now with this information, we can recall something known as Newton’s first law of motion. Newton’s first law of motion tells us that an object at rest remains at rest and an object moving with a constant velocity continues to travel at that velocity unless an unbalanced force acts on it. So, basically, what this law is telling us in relation to our system here is that if an object is at rest, then all of the forces on that object are balanced.
And in this situation, we’re not just talking about this object here with the weight 𝑊, but we can actually apply this to the entire system. Since we know from the question statement that the spring is at rest, and as a result the string and the object must also be at rest. And so, the idea is that all of the forces on the entire system, as well as the spring, the string, and the object, must be balanced.
With that in mind, let’s take a bit more of a zoomed-in view of our system, or at least the part of the system where all three objects are interacting. So, in orange here, we’ve drawn the object, in pink, is the string, and in blue, is the bottom part of the spring. Now first things first, we’ve been told that the object has a weight of 20 newtons. And we know that an object’s weight acts in a downward direction. So, we’ll label that force as 𝑊 because that’s what it’s been called in the question.
But then, as we just saw, because this object is at rest, this means that the forces on this object must be balanced. Therefore, there must be an equal force in the upward direction that is exerted by the string onto the object because the string is what’s in contact with the object. And this upward force must also have a magnitude of 𝑊 because only an upward force of magnitude 𝑊 will balance the downward force of magnitude 𝑊.
Now it’s at this point that we can recall something known as Newton’s third law of motion. Newton’s third law tells us that if an object A exerts a force on another object, object B, then B exerts an equal and opposite force on object A. So, in this situation, if we take object A to be the string and object B to be our object, then we’ve just seen that the string is exerting an upward force of magnitude 𝑊 onto our object. Therefore, by Newton’s third law of motion, our object will exert an equal force that is equal in magnitude but in the opposite direction onto the string. And therefore, we can say that on the string there is a downward force of magnitude 𝑊 acting. And that force is exerted by the object.
And it’s at this point that our diagram can get a little bit confusing. But we have to remember that this force drawn in orange and this force drawn in black are acting on the object, whereas this downward force 𝑊 is acting on the string. And with that in mind, and once again, using Newton’s first law of motion, we can see that if there is a downward force of 𝑊 acting on the string, then there must also be an upward force of 𝑊 acting on the string. Because this way the forces on the string will be balanced as well.
And so, we can draw an upward force of 𝑊 acting on the string. And that force will actually be provided by the spring. But then, if we’ve just found out that the spring is exerting an upward force of 𝑊 onto the string, then using Newton’s third law of motion once again, we can see that there must be a downward force of 𝑊 acting on the spring this time. And of course, that’s the force exerted by the string onto the spring. And of course, in this question, we are going to have to be very careful because the words string and spring sound similar.
But then, thinking about completing our diagram, we can now realise that in order for the forces on the spring to be balanced, there must also be an upward force at the top of the spring with a magnitude 𝑊, surprise, surprise. And that force is exerted by the ceiling, or whatever, the spring is attached to. But that’s not so relevant here.
So, to recap, what we have is a weight force 𝑊 acting downward on the object balanced by an upward force 𝑊 acting on the object that is exerted by the string. But then, because the string exerts an upward force of 𝑊 onto the object, by Newton’s third law, the object exerts a downward force 𝑊 onto the string. Now that downward force 𝑊 must be balanced by an upward force 𝑊 because the string is not moving. And that object is exerted by the spring onto the string.
And then, once again, using Newton’s third law of motion, we, therefore, have a downward force of 𝑊 exerted by the string onto the spring. And in order for the spring to be balanced, very finally, we’ve got an upward force 𝑊 exerted by the ceiling onto the spring.
So based on all of this information, we can say that the vertically downward force applied by the string to the spring, so that’s this force here, has a magnitude of 𝑊 which happens to be 20 newtons. And so, that’s our answer to the first part of the question.
Then we can see that the vertically upward force exerted by the spring to the string, so that’s this vertically upward force here, has also got a magnitude of 𝑊 which is 20 newtons. And then, finally, we can say that the vertically upward force exerted by the string onto the object, so that’s this vertically upward force here, is also 20 newtons. And at this point, we’ve found the answer to all our questions.