# Newton's Third Law and Forces in an Elevator

Hi,

I am confused on a certain part of Newton's third law. I know that it states that for every action force there is an equal and opposite reaction force that act of different objects. So, let's say you have an elevator that is suspended by a cable. The elevator's mass is 1kg (I know this is unrealistic but just to make the math easy), so when the elevator is sitting still, the opposing force is -9.8N and the applied force or the tension is 9.8N. There is no net force.
But let's say the elevator's motor suddenly turns on and all of a sudden the elevator and rope( the system) begin to accelerate at 2m/s^2. The net force is 2N (1kg*2m/s^2). In other words, the applied force or the tension in the rope is 3N and the opposing force or the weight is -1N.

Here is my question and although it may sound dumb, please help me understand: when I see this problem the tension in the rope attached to the elevator is 3N. But if the rope is pulling the elevator with three newtons of force, does Newton's third law not say that the elevator must pull down on the rope with 3N of force as the opposite and equal reaction force? If this is the case, why is the rope able to accelerate upward if it is being pulled down with 3N of force.

Another way to look at it is that the motor is pulling with a force of 3N (I think) so the rope must be pulling on the motor with -3N of force. Why can the motor accelerate the rope?

Thank you for all of your help!!

## Answers and Replies

Dale
Mentor
2020 Award
An ideal rope is massless so it doesn't take any net force to accelerate it. In reality the rope has some small mass so in order to accelerate there must be a small net force on the rope. E.g. if the rope is exerting 3 N on the elevator then the motor might be exerting 3.00001 N on the rope. But usually we just neglect that extra .00001 N for convenience.

Thank you for your quick response!

I am still a bit confused in that if in real life the rope was exerting 3N, then the motor might be applying 3.0000001N. Does this mean the net force is only 0.0000001N because the elevator is pulling down on the rope?

How does this apply to the massless rope?

Dale
Mentor
2020 Award
I am still a bit confused in that if in real life the rope was exerting 3N, then the motor might be applying 3.0000001N. Does this mean the net force is only 0.0000001N because the elevator is pulling down on the rope?
Yes, the net force on the rope is only 0.0000001 N. The best way to see this is to draw two free-body diagrams, one for the elevator and one for the rope. For the rope, use a mass of .000001 kg.

How does this apply to the massless rope?
Take the free-body diagram you drew above, replace the .000001 kg with an arbitrary value, m, and take the limit as m goes to 0.

This strange and slightly confusing because in all of my problems for homework so far, if the tension in the rope was 3N, then that was the applied force. But this does not seem to be the case with your explanation? How can the object still accelerate up at 2m/s^s if the net force is only .0000001N?

Dale
Mentor
2020 Award
Please do the free-body diagrams. That will answer your questions.

Dale Spam thank you for your comments and I understand if you do not want to help me anymore. This problem is simply stumping me. Whenever I draw the freebody diagram for the massless rope or the elevator, there is a 3N for action up on the elevator and a 3N force action directly down on the rope. I am so confused. How can the rope possibly accelerate up at 2m/s^2 with a 3N force downward? I know that at the point the rope connects to the motor, the motor is pulling up on the rope with 3N and the rope is pulling on the motor down with 3N. Everything seems to balance to me and I do not see any room for acceleration.

I am confused. I know you have tried and I thank you for that

Dale
Mentor
2020 Award
How can the rope possibly accelerate up at 2m/s^2 with a 3N force downward?
According to Newton's second law, how much net force is required for the rope to accelerate up at 2 m/s²? According to your free-body diagram, what is the net force on the rope?

Asking you to do the free-body diagrams is not a way to stop helping you, it is the most effective way to help you.

According to Newton's 2nd law, the rope has no mass, so there does not need to be a net force?
And by looking at my freebody diagram, the tension in the rope is 3N. 3N of applied force is pulling the elevator up and 3N is pulling the rope down, but the rope is also being pulled up by the motor at the top by 3N, so the net force in the rope is 0N?

Wait I might be understanding throught my freebody diagram. Since the 3N of the elevator pulling down on the rope and the motor pulling up on the rope have a net force of 0N and you do not need a net force to accelerate a massless rope, is this why the elevator and rope are able to accelearte at 2m/s^2 with a 2N net force (3N applied force of tension and -1N of opposing force or weight)?

Dale
Mentor
2020 Award
According to Newton's 2nd law, the rope has no mass, so there does not need to be a net force?
Correct.

And by looking at my freebody diagram, the tension in the rope is 3N. 3N of applied force is pulling the elevator up and 3N is pulling the rope down, but the rope is also being pulled up by the motor at the top by 3N, so the net force in the rope is 0N?
Yes.

Wait I might be understanding throught my freebody diagram. Since the 3N of the elevator pulling down on the rope and the motor pulling up on the rope have a net force of 0N and you do not need a net force to accelerate a massless rope, is this why the elevator and rope are able to accelearte at 2m/s^2 with a 2N net force (3N applied force of tension and -1N of opposing force or weight)?
Again, correct. This is why free body diagrams are so important. It seems like you get it now.