F = ma 2011 Exam #9 (Ideal Spring problem)

[SignaturePF]

Homework Statement

9. A spring has an equilibrium length of 2.0 meters and a spring constant of 10 Newtons/meter. Alice is pulling on
one end of the spring with a force of 3.0 Newtons. Bob is pulling on the opposite end of the spring with a force of
3.0 Newtons, in the opposite direction. What is the resulting length of the spring?
(A) 1.7 m
(B) 2.0 m
(C) 2.3 m
(D) 2.6 m
(E) 8.0 m

Homework Equations

F = - kx

The Attempt at a Solution

I just don't understand why it isn't just 2.6m.
From F = 3N and K = 10N/m,
We find that x = .3 m, so each person pulls the string .3 m in his/her direction.
Why isn't the answer 2.6 then?

 

[tms]

Take the same spring, hang it from the ceiling, and hang a 3N weight from it. What are the forces on the spring?

 

[SignaturePF]

You have 3N downward force, and - kx upward.
At the furthest point, it will be .3 m from the equilibrium point.

In the original problem:
We have 3N to the right, 3N to the left so the net force is zero?

But the answer is 2.3 m?

 

[tms]

You have 3N downward force, and - kx upward.

And what is the magnitude of the upward force?

 

[SignaturePF]

It's 3N correct? F = -10x, and if F_s = F_g, then x = .3m

 

[tms]

It's 3N correct?

You tell me. Is the spring accelerating? What does that tell you about the net force?

Going back to the original problem, what would happen if one person were pulling with 3N and the other with 4N?

 

[SignaturePF]

Well if you just have a mass hanging from the ceiling by a spring it'll undergo SHM. So at the bottom-most point the force of the spring should be greater than the force of gravity.
If one person were pulling with 4N and the other 3N, the net force would be 1N and the displacement would then be 1 = kx = .1m

 

[Simon Bridge]

Hmmm... reminds me of a gedenken physiks problem:

A string is being pulled in opposite directions by a force of 10N.
What is the tension in the string?

A. 10N
B. 20N
C: 0N
D: none of the above

Since the spring is not moving, you could nail it to the ground anywhere along it's length and get the same result.

In tms's example - the weight is pulling the spring down with 3N, and the spring pulls up with 3N. You'd have no problem computing the extension needed to do that. But what is the force that the ceiling pulls on the spring?

Ultimately, the secret is to do a free-body diagram: isolate the endpoints.

 

[tms]

Well if you just have a mass hanging from the ceiling by a spring it'll undergo SHM.

Not necessarily; it could be carefully placed at the equilibrium position.

If one person were pulling with 4N and the other 3N, the net force would be 1N and the displacement would then be 1 = kx = .1m

Not exactly. What happens to any object subjected to a non-zero net force?

 

[SignaturePF]

It experiences a net acceleration?

 

[tms]

Of course. I brought it up to point out the difference between such a situation and that in the problem, where the forces on each end are equal and there is thus no acceleration.

 

[Simon Bridge]

Note: in F=kx, x is the total extension of the spring.
Alice pulls at 3N
The spring pulls back at 3N=kx.

 

[morrisj753]

When Bob pulls with 3 N, the spring exerts a force of - 3 N to oppose it. So you account for 3N only once.
F=kx
3 = 10x
x = 0.3 m
2 + 0.3 = 2.3 m

 

[Simon Bridge]

When Bob pulls with 3 N, the spring exerts a force of - 3 N to oppose it. So you account for 3N only once.

That's right.
How far does Bob move?

Compare:
Replace Alice with a wall.
When bob pulls on the spring with 3N, how hard is the wall pulling on the spring?

 

[SignaturePF]

To answer your questions Simon:
Bob moves .3m (from F = -kx)
The wall has to pull 3N to the left in order to keep the spring from being ripped off the wall.
Oh, that makes sense; Alice only functions to keep the spring from being let go from her hands. Her 3N force goes to keeping the spring's left end in place. Since the spring is being pulled 3N, Alice, with her 3N force, keeps the spring fixed. Is that right?

 

[Simon Bridge]

Oh, that makes sense; Alice only functions to keep the spring from being let go from her hands. Her 3N force goes to keeping the spring's left end in place. Since the spring is being pulled 3N, Alice, with her 3N force, keeps the spring fixed. Is that right?

Close - but doesn't exactly the same argument apply to Bob? vis:

Bob only functions to keep the spring from being let go from his hands. His 3N force goes to keeping the spring's left end in place. Since the spring is being pulled 3N, Bob, with his 3N force, keeps the spring fixed.

So which is it?
You have an extra bit of information - the spring com is stationary.
So this situation is the same to each person as if they were pulling on a half-length spring fastened to a wall.