# Calculate this Spring's Potential Energy

#### EEristavi

Homework Statement
If we have object with m = 1 hanging on a spring and elongation is h = 0.02.
What is the potential energy of the spring after it's being stretched?
Homework Equations
E = mgh
I know that gravitational potential energy is decreased by E = -m g h = -1 10 0.02 = -0.2. So, the spring potential energy must be E=0.2 (Joule).
However, in the answer's sheet I have E=0.1

What mistake do I make?

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#### PhanthomJay

Homework Helper
Gold Member
The elongation is apparently 0.02 m when the mass is held by your hand and slowly allowed to reach its rest position when you release your hand. Your hand does work in this case which you have neglected. Or alternatively, if you release the mass at h = 0 , it will have kinetic energy as well as it passes h= 0.02.
Look at another approach.

#### David Lewis

I'm going to assume m = 1 kg, h = 0.02 m, and g = 10 m/s2. The force that the spring exerts on the weight is not constant. It depends on how far the spring is stretched.

#### kuruman

Homework Helper
Gold Member
The question is ill posed. The zero of potential energy can be chosen at will. Without specifying where the zero is, the question cannot be answered. To be specific about the elastic energy, in a horizontal spring-mass system the zero is normally chosen at the relaxed position of the spring; in a vertical spring-mass system the zero is normally chosen at the equilibrium position where the net force on the hanging mass is zero.

In this question we are asked to find the potential energy of the spring, not the mass, so mgh is not part of the answer.

#### archaic

Elastic potential energy is not the opposite of gravitational potential energy. In the case of vertical springs, however, the weight of the object and the elastic force are.
Recall that $|\vec F_s|=kx=mg=|\vec F_g|$, that $U_s=\frac{1}{2}kx^2$ and that $U_g=-mgx$ where $x$ is the elongation of the spring. Try to find a relation between both potential energies.
Hint : $\frac{U_g}{U_s} = ?$
NB : here we choose $U_g=-mg\int_{h_i}^{h_f}\mathrm{dh}$.

#### EEristavi

I think you are making mistake here.
If hand doesnt move - displacement is 0.
A = F S = F * 0 = 0

#### EEristavi

I'm going to assume m = 1 kg, h = 0.02 m, and g = 10 m/s2. The force that the spring exerts on the weight is not constant. It depends on how far the spring is stretched.

Spring is stretched by h = 0.02 or $\Delta x = 0.02$

#### EEristavi

The question is ill posed. The zero of potential energy can be chosen at will. Without specifying where the zero is, the question cannot be answered. To be specific about the elastic energy, in a horizontal spring-mass system the zero is normally chosen at the relaxed position of the spring; in a vertical spring-mass system the zero is normally chosen at the equilibrium position where the net force on the hanging mass is zero.

In this question we are asked to find the potential energy of the spring, not the mass, so mgh is not part of the answer.
Maybe I translated poorly.
I'm asking what is the potential energy of the spring After it's being stretched (I will change it in the problem statement also, thanks).

So, I'm just calculating the change of potential energy - so it doesn't really matter where I take zero

#### EEristavi

Guys it's Introductory physics homework - it should be easy, don't overthink please :)
I'm just missing something really easy

#### archaic

Guys it's Introductory physics homework - it should be easy, don't overthink please :)
I'm just missing something really easy
If you follow the advice I gave you, you'll find that $U_g=-2U_s$, you only need to plug in what you have now. Try to do it yourself, though.
Another hint : $kx^2=kxx$, $|\vec F_g|=|\vec F_s|$.
Elastic potential energy is not the opposite of gravitational potential energy. In the case of vertical springs, however, the weight of the object and the elastic force are.
Recall that $|\vec F_s|=kx=mg=|\vec F_g|$, that $U_s=\frac{1}{2}kx^2$ and that $U_g=-mgx$ where $x$ is the elongation of the spring. Try to find a relation between both potential energies.
Hint : $\frac{U_g}{U_s} = ?$
NB : here we choose $U_g=-mg\int_{h_i}^{h_f}\mathrm{dh}$.

#### PhanthomJay

Homework Helper
Gold Member
I think you are making mistake here.
If hand doesnt move - displacement is 0.
A = F S = F * 0 = 0
As noted by kuruman, the question is ill posed. I am assuming that when the mass is first placed onto the hanging spring , it is held there by your hand and slowly lowered to the point at 0.02 m where when you release your hand, the mass doesn’t move and the spring stops stretching. Your hand does work because it is displaced 0.02 m in this process. So you can’t say that spring final PE is equal to initial gravitational PE because other work is done.
You should explore the fact that if the mass is suddenly released by your hand at the very start and you immediately let go without doing work, that the spring will displace 0.04 m before it stops and then rise up again and down again until ultimately it is damped out and settles at the 0.02 m mark.

#### EEristavi

Your hand does work because it is displaced 0.02 m in this process. So you can’t say that spring final PE is equal to initial gravitational PE because other work is done.
This is the situation when you move your reference point or potential's zero with mass.
My reference point was not changing with respect to "Earth".
These are 2 different points of view and both are correct.

Hope you understood what i'm saying.

So, This doesn't mean that my approach isn't correct.

#### EEristavi

$Ug=−2Us$
Ok. Now let me think: why do we have times 2 with Spring potential

#### EEristavi

If you follow the advice I gave you, you'll find that $U_g=-2U_s$, you only need to plug in what you have now. Try to do it yourself, though.
Another hint : $kx^2=kxx$, $|\vec F_g|=|\vec F_s|$.
I now that if my answer is divided by 2 - it's correct.
I cant understand why there is 2....

#### archaic

Our variables : $|\vec F_s|=kx=mg=|\vec F_g|$, $U_s=\frac{1}{2}kx^2$ and $U_g=-mgx$.
$$\frac{U_g}{U_s}=\frac{-mgx}{\frac{1}{2}kx^2}=\frac{-mgx}{\frac{1}{2}kxx}$$
From the first equation we have $kx=mg$.
$$\frac{U_g}{U_s}=\frac{-mgx}{\frac{1}{2}(kx)x}=\frac{-mgx}{\frac{1}{2}mgx}=-\frac{1}{\frac{1}{2}}=-2$$
Or, in other words, gravitational potential energy $U_g=-2U_s$ elastic (spring's) potential energy.

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#### EEristavi

Woow...
now I get it!!!!!

Very nice!
Now I have to think what's the physics behind it (this I will manage on my own).. :D

Thank you very much!

#### kuruman

Homework Helper
Gold Member
Woow...
now I get it!!!!!

Very nice!
Now I have to think what's the physics behind it (this I will manage on my own).. :D

Thank you very much!
You have to understand that the ratio of -2 is not always the case but only if the zero of potential gravitational energy is taken at the same point as the zero of elastic energy. In general,
$$\frac{U_g}{U_s}=-\frac{mg(x-x_{0g})}{\frac{1}{2}k{\left(x^2-x_{0s}^2\right)}}$$where $x_{0g}$ and $x_{0s}$ are, respectively, the points where the gravitational and spring potential energy are zero. The ratio is -2 only if $x_{0g}=x_{0s}=0$.

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#### archaic

You have to understand that the ratio of -2 is not always the case but only if the zero of potential gravitational energy is taken at the same point as the zero of elastic energy. In general,
$$\frac{U_g}{U_s}=\frac{mg(x-x_{0g})}{\frac{1}{2}k{\left(x^2-x_{0s}^2\right)}}$$where $x_{0g}$ and $x_{0s}$ are, respectively, the points where the gravitational and spring potential energy are zero. The ratio is -2 only if $x_{0g}=x_{0s}=0$.
In my example, $x$ is the elongation, I should've written it as
You have to understand that the ratio of -2 is not always the case but only if the zero of potential gravitational energy is taken at the same point as the zero of elastic energy. In general,
$$\frac{U_g}{U_s}=\frac{mg(x-x_{0g})}{\frac{1}{2}k{\left(x^2-x_{0s}^2\right)}}$$where $x_{0g}$ and $x_{0s}$ are, respectively, the points where the gravitational and spring potential energy are zero. The ratio is -2 only if $x_{0g}=x_{0s}=0$.
Isn't the general formula for EPE $U_s=\int_{l_0}^lk(x-l_0)dx=\frac{k(l^2-l_0^2)}{2}-kl_0(l-l_0)$ if we are taking points not displacement? I think you should specify that you're taking the spring's $l_0$ point as $x=0$.

#### jbriggs444

Homework Helper
I cant understand why there is 2....
That 2 is the same as the 2 in the formula for the area of a triangle: a = 1/2 base * height -- average width times total height. The work done against the spring as it elongates to its final extension varies from zero to the full force. The average force is equal to 1/2 the full force. The work done against the spring is equal to the average force times the distance extended.

It is also the same as the 2 in $\int x\ dx = \frac{1}{2}x^2$

Which is, by no coincidence, the same as the 2 in the formula for potential energy of a spring, $E=\frac{1}{2}kx^2$

#### EEristavi

Guys I understood it mathematically. However, I cant figure it out as "Change of energy concept" -
If the change of potential energy is $mg \Delta x$, why the spring's potential energy isn't increased the same amount. As I understand we have a closed system (Or Isn't it?.....).

This is the root of my mistake

#### kuruman

Homework Helper
Gold Member
Are you asking why mechanical energy is not conserved? To examine mechanical energy conservation, you need to have mass $m$ move from point A to point B. What are points A and B in this case, how does the mass move from A to B and what is the mechanical energy at each of the two points?

#### EEristavi

I have 2 Questions:

1. is mechanical energy conserved?
2. If it's not, why?

Note:
As I see it's not conserved. Am I right?

#### kuruman

Homework Helper
Gold Member
I have 2 Questions:

1. is mechanical energy conserved?
2. If it's not, why?

Note:
As I see it's not conserved. Am I right?

#### EEristavi

What are points A and B in this case
point A - where spring isn't stretched yet.
Point B - where spring is stretched by $\Delta x$

how does the mass move from A to B
It moves because of the gravity.
other force that is involved is force from spring F = k x

what is the mechanical energy at each of the two points
if we take "starting point or 0 (zero)" to be point A:
Mechanical Energy at point A - 0;
Mechanical Energy at point B - $mg \Delta x$
P.S. one way or another - Change of energy will always be $mg \Delta x$

#### kuruman

Homework Helper
Gold Member
OK, that's good, but is it held by a hand and then released or is it gently lowered by the hand until it reaches the equilibrium position?

"Calculate this Spring's Potential Energy"

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