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

• SignaturePF
In summary, the conversation discusses the forces and displacement involved in a scenario where two individuals, Alice and Bob, are pulling on opposite ends of a spring with a force of 3N each. The spring has an equilibrium length of 2.0 meters and a spring constant of 10 Newtons/meter. Through the use of free-body diagrams and the equation F = -kx, it is determined that the resulting length of the spring is 2.3 meters, with each person pulling the spring 0.3 meters in their respective directions. The conversation also explores the concept of a stationary spring center of mass and how it relates to the forces exerted by Alice and Bob.
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

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?

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

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?

SignaturePF said:
You have 3N downward force, and - kx upward.
And what is the magnitude of the upward force?

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

SignaturePF said:
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?

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

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.

SignaturePF said:
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?

It experiences a net acceleration?

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.

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

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

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?

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?

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.

## 1. What is the equation F = ma used for?

The equation F = ma is used to calculate the force (F) exerted on an object by multiplying its mass (m) by its acceleration (a). This is known as Newton's Second Law of Motion.

## 2. What is the 2011 Exam #9 Ideal Spring problem?

The 2011 Exam #9 Ideal Spring problem is a physics question that involves calculating the force exerted by a spring on a mass based on its displacement from its equilibrium position.

## 3. How do you solve the Ideal Spring problem?

To solve the Ideal Spring problem, you will need to use the formula F = -kx, where F is the force, k is the spring constant, and x is the displacement from equilibrium. You will also need to know the mass of the object and its acceleration, which can be calculated using F = ma.

## 4. What is the significance of the negative sign in the formula F = -kx?

The negative sign in the formula F = -kx indicates that the force exerted by the spring is opposite to the direction of the displacement. This means that the force will always try to bring the object back to its equilibrium position.

## 5. Can the Ideal Spring problem be applied in real-life situations?

Yes, the Ideal Spring problem can be applied in real-life situations, such as in the design of shock absorbers for cars or in the calculation of forces exerted by springs in various mechanical systems.

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