What happens if we apply different forces in opposite directions on both ends of a spring?

If I have a spring and I apply force F1 towards the right on the right end and F2 towards the left on the left end, what will happen? There are no blocks, just a light spring.

How can I find the extension of the spring in this case?

• When you say that the spring is "light", do you mean that is has no mass? – sammy gerbil Sep 14 '18 at 0:41

Draw a free body diagram and look at all the forces on the spring.

You can break it down into essentially 3 forces. You can have the two equal in magnitude but opposite direction forces to determine how much the spring will extend.

You can also see that those forces will cancel, leaving a net force acting on the spring which is not capable of extending the spring; and instead moves it like it would any object with an applied net force.

• OK thanks... so will the force used to stretch the spring be (F1 + F2)/2? – Ankit Kumar Misra Sep 14 '18 at 2:02
• Try to think about how you would usually see a force on a spring. If you can separate it into two equal but opposite forces and a remaining net force, you should see what happens. – JMac Sep 14 '18 at 13:23

What will happen? - The spring will stretch.

How can I find the extension of the spring in this case? - Linear springs (I'm assuming yours is linear) obey Hook's law: F = -k*x. If you want to know how much force you need stretch the spring 5 cm (for example), you need to know the spring constant k. If you want to find the spring constant you need to measure how much force did you apply and how much did the spring stretch. If you want to know how much did the spring stretch, you need to know how much force did you applied and the spring constant.

Edit: The above wrongly assumed that F1 = F2. If F1 and F2 are different in magnitude then you wont have a system in equilibrium and you would effectively displacing the (already stretched spring) in the direction with the greater force.

• The title of the question mentions two different forces acting on the spring. I think they are talking about a situation where $F_1 \neq F_2$. – JMac Sep 13 '18 at 19:26
• Oh, I missed that while writing the answer... I'll edit accordingly. – user190081 Sep 13 '18 at 19:29