# Stretching of string (and tearing of string)

#### DesertFox

Hello everbody!

I found the following representation: F = 2ρdl(d²l/dt²)

F - the maximum stretching force (force at which the string tears);
ρ - density;
dl - random differential section of the string;
t - time.

This is some kind of parallel between WAVE and STRING? (Dull)(Headbang)

I cannot find information in the world biggest search machine, so every single explaining comment will be highly appreciated...

Thank you!!!(Cool)

#### studiot

Well I don't recognise your equation, where does it come from?

Force has dimensions

$$\displaystyle M{L^{ + 1}}{T^{ - 2}}$$

but the right hand side of your equation has different dimensions since for

$$\displaystyle \rho (dL)\frac{{{d^2}L}}{{d{t^2}}}$$

the dimensions are

$$\displaystyle \left( {M{L^{ - 3}}} \right)\left( L \right)\left( {L{T^{ - 2}}} \right) = M{L^{ - 1}}{T^{ - 2}}$$

#### DesertFox

Hmm..
What do the different dimensions in the equation (which i quoted) imply?

#### studiot

Well the missing quantity is L squared or area.

So this must be acknowledging the fact that as you stretch a string it gets thinner to make your equation work.

Sorry the mathml doesn't seem to have taken on this forum. It used to work.

#### DesertFox

So, the equation is right?

Then, it is confusing why i cannot find information about it in the textbooks and google... I cannot grasp its derivation.

#### studiot

No the equation must be missing a term equivalent to area.

Density = mass / volume so the equation is basically Newton's Second Law when multiplied by area since the second derivative is acceleration.

An alternative would be to transfer the area to the left hand side so it becomes Force / area or stress.

#### DesertFox

In the original text, it is mentioned for real that the equation has something to do with Newton's Second Law... but no more explanation.

Maybe things are getting more clear for me... But..

Could you, please, write down the equation in the correct way (acknowledging the area)...

#### DesertFox

by the way:

"ρ" is described as "mechanical" density. I have never heard of "mechanical density" and i don't understand what does it mean in the context of stretching (tearing) of a string.

#### ChipB

PHF Helper

$$\displaystyle F = \rho A dl (d^2l/dt^2)$$

where A is the cross-sectional area of the string. This equation is simply a form of F=ma. It has nothing to do with waves, but rather is an equation that determines how a string stretches dynamically assuming it has zero stiffness (i.e. its spring constant = 0).

I suspect that in the equation you gave p, density, is in units of Kg per meter of length of the string, not Kg per cubic meter which is how it's normally defined. With this understanding the units work out, but I have no idea why there is a factor of 2 in your equation.

#### studiot

Consider a differential length of string, dL, with cross sectional area A.

The force in the string = stress x A or stress = Force/A

The mass of the differential length = density x volume = density x dL x A

So, by Newton's Second Law

Force = mass x acceleration = density x dL x A x acceleration.

So stress in the string = Force/A = F/A = density x dL x acceleration.

But force does not break a string, stress does.

The string will break when the maximum stress reaches the breaking stress.

I think the factor of 2 is to do with the fact that the maximum stress (or load) is twice the average stress (or load), but it is not clear from your translation what force (load) or stress we are talking about.