Potential energy of rubber bands

[rasalzari]

<Mentor's note: moved from a general forum and therefore no template>

This is the equation of potential energy in rubber bands, however, I don't know how to calculate the inertia of the arm of catapults or get the k constant. And what does n mean? Basically, i don't understand the whole equation.
The energy balance, where N is the number of rubber bands, k is the spring constant of a single rubber band, x0 is the unstretched length of a rubber band, m the mass of the ball, and I is the moment of inertia of the arm.
Can someone please show me an example of how to calculate this?
I used a marshmallow which is 0.007 kg, I stretched the catapults to 0.1 m, it has an angle of 35 degrees
range=0.793
initial velocity=2.88 m/s
time=0.337 s
max height=0.139 m
horizontal velocity=2.36 m/s
vertical velocity=1.65 m/s
Thats all i know and i hope it helps! please help me figure this out!

 

[haruspex]

<Mentor's note: moved from a general forum and therefore no template>

View attachment 150616
This is the equation of potential energy in rubber bands, however, I don't know how to calculate the inertia of the arm of catapults or get the k constant. And what does n mean? Basically, i don't understand the whole equation.
The energy balance, where N is the number of rubber bands, k is the spring constant of a single rubber band, x0 is the unstretched length of a rubber band, m the mass of the ball, and I is the moment of inertia of the arm.
Can someone please show me an example of how to calculate this?
I used a marshmallow which is 0.007 kg, I stretched the catapults to 0.1 m, it has an angle of 35 degrees
range=0.793
initial velocity=2.88 m/s
time=0.337 s
max height=0.139 m
horizontal velocity=2.36 m/s
vertical velocity=1.65 m/s
Thats all i know and i hope it helps! please help me figure this out!

That equation is rather unusual. It implies the force needed to stretch the rubber rises as the square root of the extension, whereas the usual law says it rises linearly. Is it possible that x here refers not to the actual extension but to some other displacement within a mechanism supporting the rubber band? Is there a diagram?

 

[rasalzari]

That equation is rather unusual. It implies the force needed to stretch the rubber rises as the square root of the extension, whereas the usual law says it rises linearly. Is it possible that x here refers not to the actual extension but to some other displacement within a mechanism supporting the rubber band? Is there a diagram?

Yes here you go

And here is a link with further information
http://www.sciencebuddies.org/content/pdfs/projectideaskits/phys_p089/phys_p089_20131021.pdf

 

[haruspex]

Ok, I see that the text states that the rubber bands provided do have a nonlinear force/extension relationship.

I don't know how to calculate the inertia of the arm

It gives a formula in note 3 under Experimental Procedure.
However, I see that it defines I as the moment inertia about the mass centre of the arm. That is not what you want. The arm rotates about one end, not its mass centre. The text says to use I = 1.1 x 10^-3kg*m². If that really is the MoI about the mass centre then you need to add ML²/4 where L is the length of the arm and M its mass. But it is possible they mean the MoI about one end is 1.1 x 10^-3kg*m².
You can check this by measuring L and M and using:
MoI about mass centre = ML²/12
MoI about one end = ML²/12+ML²/4=ML²/3.
See which of those is closer to 1.1 x 10^-3kg*m².

get the k constant

See note 5 under Experimental Procedure.

what does n mean?

As you wrote, N is the number of bands in parallel. I think it is up to you how many you attach.

 

[rasalzari]

Ok, I see that the text states that the rubber bands provided do have a nonlinear force/extension relationship.

It gives a formula in note 3 under Experimental Procedure.
However, I see that it defines I as the moment inertia about the mass centre of the arm. That is not what you want. The arm rotates about one end, not its mass centre. The text says to use I = 1.1 x 10^-3kg*m². If that really is the MoI about the mass centre then you need to add ML²/4 where L is the length of the arm and M its mass. But it is possible they mean the MoI about one end is 1.1 x 10^-3kg*m².
You can check this by measuring L and M and using:
MoI about mass centre = ML²/12
MoI about one end = ML²/12+ML²/4=ML²/3.
See which of those is closer to 1.1 x 10^-3kg*m².

See note 5 under Experimental Procedure.

As you wrote, N is the number of bands in parallel. I think it is up to you how many you attach.

Ohh i see thank you so much for your reply, however it only applies for non-linear? but my graph is linear... is there any other way to get the potential energy for linear relationship?

 

[haruspex]

my graph is linear

What graph? What values are you plotting on the axes?

 

[rasalzari]

What graph? What values are you plotting on the axes?

 

[haruspex]

View attachment 151849

Interesting.
What are the values of x₂-x₀ for each band?

 

[rasalzari]

Interesting.
What are the values of x₂-x₀ for each band?

Oh sorry I am a newbie in physics and math, could you explain what x2-x0 mean?

 

[haruspex]

Oh sorry I am a newbie in physics and math, could you explain what x2-x0 mean?

They appear in the equation you posted originally. They are defined in the link you provided. I have the feeling you have not read that page fully.

 

[rasalzari]

They appear in the equation you posted originally. They are defined in the link you provided. I have the feeling you have not read that page fully.

no, I did but I didn't cover much of physics so i couldn't understand that's why i asked in this website so i can learn more

 

[haruspex]

no, I did but I didn't cover much of physics so i couldn't understand that's why i asked in this website so i can learn more

This much is not a question of physics. What does it say x₀, x₁ and x₂ represent?

 

[rasalzari]

This much is not a question of physics. What does it say x₀, x₁ and x₂ represent?

I don't really know but at first, I thought it might mean initial velocity, is it?

 

[haruspex]

I don't really know but at first, I thought it might mean initial velocity, is it?

No, they're lengths of the elastic. As it says, x₀ is the unstretched length. I thought it defined x₁ and x₂, but maybe I imagined it. x₁ is the stretched length just before firing. At the end of firing (release of projectile) it may not have returned to its unstretched length, instead being length x₂.

 

[rasalzari]

No, they're lengths of the elastic. As it says, x₀ is the unstretched length. I thought it defined x₁ and x₂, but maybe I imagined it. x₁ is the stretched length just before firing. At the end of firing (release of projectile) it may not have returned to its unstretched length, instead being length x₂.

ohhh alright thank you sir! i will read more about the topic and if i have more questions ill ask

 

[Dr.D]

In general, rubber is nonlinear and nonconservative. This means that
(1) force will not be proportional to the elongation;
(2) work put in will not be fully recovered when the stretched piece of rubber is released.
This makes energy calculations with rubber bands rather iffy.

 

[kuruman]

ber is nonlinear and nonconservative. This means that
(1) force will not be proportional to the elongation;
(2) work put in will not be fully recovered when the stretched piece of rubber is released.

To which I will add that rubber bands also display hysteresis; the work that is recovered depends on what has already been done to the band in terms of stretching and relaxing.