# I Stretched spring and change in mass

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1. Nov 11, 2016

### Mohammed Shoaib

I came across with this question in my work.
A stretched spring has greater energy and therefore greater mass than an unstretched spring. What's the mass increase when you stretch a spring with 500N/m by 40cm?

My question is How mass can increase with stretching of the spring?
As per fundamental physics mass is conserved. I am confused. Kindly help.

2. Nov 11, 2016

### Staff: Mentor

The relationship between energy, mass, and momentum is $m^2 c^2=E^2/c^2-p^2$

Mass is only conserved for isolated systems. A spring cannot be stretched if it is isolated.

3. Nov 12, 2016

### Mohammed Shoaib

But sorry
I did not get that?

Why did you use relativistic energy momentum for a mass - spring mass system.

4. Nov 12, 2016

### haushofer

Because you ask how mass increases if Energy increases . Newton cannot answer that question because mass and energy are separately conserved in classical mechanics .

5. Nov 12, 2016

### Staff: Mentor

Because it is the only theory where this claim is true:
If you did not intend to discuss relativity then you can simply neglect the whole idea as an insignificant relativistic effect.

6. Nov 12, 2016

### timmdeeg

By being stretched the spring gains potential energy, then according to $E=mc^2$ it's mass increases.

7. Nov 15, 2016

### Mohammed Shoaib

How can we relate the factor c speed of light in the E=mc2 and say this equation explains the increase in mass of a stretched spring. kindly explain the role of c here.

8. Nov 15, 2016

### Ibix

It's the unit conversion factor between mass and energy. You can set it to 1 by an appropriate choice of units if you like.

9. Nov 15, 2016

### Staff: Mentor

It is just a unit conversion factor. The SI unit system was started before relativity was understood, so whenever you are looking at relativistic quantities you will see factors of c used to convert SI units.

An analogy would be an ancient sailor measuring vertical distances in fathoms and horizontal distances in nautical miles. You would wind up with conversion factors any time you had a formula that includes both a horizontal distance and a vertical distance.

Similarly, energy and momentum are the timelike direction and the spacelike direction of the same thing. They could be measured in the same units, and when you do not then you get factors of c in the relationship $m^2 c^2 = E^2/c^2-p^2$

10. Nov 15, 2016

### Mister T

Can you first show us that you know how to solve this problem?

That is, calculate the amount the energy in joules, and then convert it to a mass in kilograms. Once you do that it may become obvious to you why it's safe to ignore the contribution made to the spring's mass by stretching the spring.

It's really necessary that you first do this so that you'll have the context needed to understand our answer to your question.

11. Nov 16, 2016

### Mohammed Shoaib

Potential energy U=1/2kx2 = 40J and m from e=mc2 gives m=4.44x10^-16kg.
Thanks for support.

12. Nov 16, 2016

### Mister T

Your calculation shows that the mass increase is ridiculously small, several orders of magnitude smaller than the mass of an single electron.

The role of $c$ in $E=mc^2$ doesn't explain the increase in mass any more than the role of $k$ in $E=\frac{1}{2}kx^2$ explains the increase in energy. There are derivations you can study that will explain the validity of equations such as $E=\frac{1}{2}kx^2$ and $E=mc^2$, and it's ultimately up to people to verify by experiment that these equations are valid. All of that is done before they appear in textbooks and we are asked to learn them.

13. Nov 16, 2016

### Staff: Mentor

It is more like the mass of a small bacterium. Still very small, but much larger than an electron or even a proton.

I agree with the remainder of your post.

14. Nov 16, 2016

### Mister T

Oops! Of course you're right. It's several orders of magnitude LARGER than an electron's mass.