How much does the spring stretch?

In summary, the conversation discusses the question of how much a spring will stretch when a mass attached to it is resting on a board that is accelerating. The solution involves using Newton's Laws of Motion and considering the forces acting on the mass and the board. After some reasoning and corrections, it is determined that the amount of stretch is dependent on the acceleration of the board and can be calculated using the equation ##x=\frac{m(g-a)}{k}##, where ##m## is the mass of the object, ##g## is the acceleration due to gravity, ##a## is the acceleration of the board, and ##k## is the spring constant.
  • #1
kaspis245
189
1

Homework Statement


A mass ##m## body is attached to the spring and rests on the board so the spring is not stretched. The board starts to move with an acceleration ##a##. How much does the spring stretch at the moment when the board separate from the body?
Picture.png

Homework Equations


Newton's Laws of Motion

The Attempt at a Solution


Here is my drawing:
new.png


I think that the board will separate from the body when ##kx=m_1a##, where ##m_1## is the mass of the board. I can't use ##m_1## since it is not given in the problem but I can't think a way to replace it.

Is my reasoning correct and how can I replace ##m_1##?
 
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  • #2
Hello Kaspis. Funny you should draw a picture with three forces and then mention only two to find the separation. And these two work on two different objects!

Think about what the forces are on the mass m. There is a third one. And remember Newton 2.
 
  • #3
Here's the corrected image (please nevermind how the forces are represented, only directions matter here):
onwPSAk.png

I know that ##F_N=mg##, so ##F_N## should be 0 when ##kx=mg##, therefore ##x=\frac{mg}{k}##. Is that the answer?

Update:
Now that I think about it, the equation ##x=\frac{mg}{k}## should not suffice. The amount of stretch is clearly dependant on the acceleration with which the board is falling. For example, if the board is falling with many times bigger acceleration than ##g## it will surely separate from the board sooner than it would with smaller acceleration than ##g##. Thus, there must be some kind of relationship between board's acceleration and other forces.
 
Last edited:
  • #4
How do you know ##F_N = mg## ? That is only when the board is at rest at the point where the extension of the spring is zero. Once there is acceleration that can't be true any more.
 
  • #5
BvU said:
How do you know ##F_N = mg## ?

Sorry, I'll apply Newton's second law here:
##kx+F_N-mg=-ma_1##, where ##a_1## is the acceleration with which the body is moving. Unfortunately, this doesn't help at all :frown:.

Maybe I need to find the rate at which ##F_N## decreases and the rate at which ##m_1a## increases. Would that help?
 
  • #6
##a## is a given acceleration. As soon as ##F_N## hits 0, the mass comes off the board. Only one unknown: ##x## !
 
  • #7
Oh, so the equation should be like this:
##kx+F_N-mg=-ma##

If ##F_N## is 0, then:
##kx-mg=-ma##
##x=\frac{m(g-a)}{k}##

Is that correct?
 
  • #8
What do you think ? Does it look plausible ?
- it goes to ##kx = mg## for ##a## goes to 0, so that seems OK
- if ##a = g## there is free fall from the start, so that looks OK too
- there are no strange outsiders contributing (such as ##m_1##), which seems good

I'm all in favour of this outcome ! Well done.
 
  • Like
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1. How does the mass affect how much a spring stretches?

The mass of an object does not directly affect how much a spring stretches. However, the weight of the object, which is determined by its mass and the force of gravity, does affect the stretching of a spring.

2. What factors contribute to how much a spring stretches?

The amount of force applied to the spring, the spring's stiffness, and the distance the spring is stretched all contribute to how much a spring stretches. Additionally, the material and thickness of the spring can also affect its stretching capability.

3. How do I calculate the amount a spring stretches?

The amount a spring stretches can be calculated using Hooke's Law, which states that the force applied to a spring is directly proportional to the distance the spring stretches. The equation is F = -kx, where F is the force applied, k is the spring constant, and x is the distance the spring stretches.

4. Can a spring stretch indefinitely?

No, a spring cannot stretch indefinitely. There is a maximum limit to how much a spring can stretch before it becomes permanently deformed or breaks. This limit is known as the elastic limit and varies depending on the material and design of the spring.

5. How can I increase the amount a spring stretches?

The amount a spring stretches can be increased by either applying more force to the spring or by using a spring with a lower stiffness. However, it is important to not exceed the spring's elastic limit to avoid permanent damage.

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