Net force an an infinitesimal string

[ehrenfest]

Homework Statement

The book I am using (Zwiebach on page 66) uses the expression

dF_v = T_0 \frac{ \partial{y}}{\partial{x}} |_{x+dx} -T_0 \frac{ \partial{y}}{\partial{x}} |_{x}

for the force on an infinitesimal length of string. We assume dy/dx is much less than 1. I am not sure how the author got these expressions for the tension at x and x+dx? Is he using a Taylor series approximation?

Homework Equations

The Attempt at a Solution

 

[Archduke]

Hey there,

First, try splitting the tension acting along the string into the x and y direction (see attachment). We know that the tension along the x direction (I've called this T_{0}) have got to cancel out, else the string will be accelerating along the x-direction. So, looking in the y-direction and using geometry, the tensions at x and x+\Delta x are:

At x : T_{(x)y} = - T_{0} \tan{\theta_{x}}

At x+\Delta x : T_{(x + \Delta x)y} = T_{0} \tan{\theta_{x + \Delta x}}

But, we know that, for small theta, \tan{\theta}\approx \frac{\partial {y}}{\partial {x}}. So:

T_{(x)y} = - T_{0} \tan{\theta_{x}} = - T_{0} \frac{\partial{y}}{\partial{x}} |_{x}

and

T_{(x + \Delta x)y} = T_{0} \tan{\theta_{x + \Delta x}} = T_{0}\frac{\partial{y}}{\partial{x}} |_{x + \Delta x}

And the net force acting on the string is just the sum of these components, or;

dF_v = T_{x + \Delta x}_{y} - T_{x}_{y} = T_0 \frac{ \partial{y}}{\partial{x}} |_{x+dx} -T_0 \frac{ \partial{y}}{\partial{x}} |_{x}

 

[ehrenfest]

Hey there,

First, try splitting the tension acting along the string into the x and y direction (see attachment). We know that the tension along the x direction (I've called this T_{0} have got to cancel out, else the string will be accelerating along the x-direction.

But I think T_0 is defined in my book as the tension along the string, not in the x direction. Don't we want sin(theta) T_0 to get the force in the y-direction, not tan(theta) T_0?

Also, T_0 is defined in my book (and I should have written this in my first post) as the tension along the string. Thus, there is a small net force in the x-direction. However, it is negligable.

Also, when you say

T_{(x)y} = - F_{0} \tan{\theta_{x}}

do you mean

T_{(x)y} = - T_{0} \tan{\theta_{x}}

?

Because the first equation seems incorrect to me?

 

[Archduke]

Also, when you say

T_{(x)y} = - F_{0} \tan{\theta_{x}}

do you mean

T_{(x)y} = - T_{0} \tan{\theta_{x}}

Yeah, sorry about that. I've changed it now.

As for the tension in the x-direction, there is some there, but it is canceled out by tension of the same magnitude, but opposite direction, at x + \Delta x.

As for the sin/tan problem, it appears we're both right, because we've chosen to call T_0 something different. However, I can't see how we've then got the same expression.

I'll be away for a few days, so hopefully someone else can resolve it for you. Sorry if I've just added to your confusion with the different notation!

 

[ehrenfest]

As for the tension in the x-direction, there is some there, but it is canceled out by tension of the same magnitude, but opposite direction, at x + \Delta x.

Actually the tension is not necessarily the same at x and x+dx because dy/dx is evaluated at each of these points and could be different.

And yes someone else please help clarify the sin/tan issue please!

 

[Dick]

Inasmuch as you've assumed |dy/dx|<<1 it doesn't matter whether you use sin or tan. They are both the same to the order you are concerned about. And the tension is the same if the string is massless. It couldn't support a variable tension w/o producing infinite accelerations.

 

[ehrenfest]

Inasmuch as you've assumed |dy/dx|<<1 it doesn't matter whether you use sin or tan. They are both the same to the order you are concerned about.

I see. Thanks.

And the tension is the same if the string is massless. It couldn't support a variable tension w/o producing infinite accelerations.

If the string is massless, it could not support a net vertical tension either. So, it does have a density.

 

[Dick]

If the string is massless, it could not support a net vertical tension either. So, it does have a density.

Even if you are dealing with massive string (and if you are doing string theory, you are usually thinking of fundamental massless strings - really). One should be able to argue that the tension variation can be neglected to the order you are working at, just like the difference between sin and cos. But I'm having a hard time coming up with an argument simple enough to be convincing.

 

[ehrenfest]

Even if you are dealing with massive string (and if you are doing string theory, you are usually thinking of fundamental massless strings - really). One should be able to argue that the tension variation can be neglected to the order you are working at, just like the difference between sin and cos. But I'm having a hard time coming up with an argument simple enough to be convincing.

Firstly, you probably mean sin and tan. Secondly, I am doing ST but I am in Zwiebach Chapter 4 which deals with nonrelativistic strings and he explicitly says that the string has a density. Thirdly, I am not sure what you mean about neglecting tension variation. The difference between the vertical tension at x and x + dx can be shown to be much larger than the difference between the horizontal and it is not negligable at least in the context of the problems I am doing.

 

[Dick]

Sure, I mean sin and tan. And I am not talking about ignoring the difference between the components of tension, I'm talking about ignoring the absolute value of the tension variation. Which I'm having problems coming up with a good argument for. Can you help?

 

[ehrenfest]

I see. You're question is why can we use T_0 at every point on the string, right?

 

[Dick]

Right. Ok. Here's my answer. In principle, you should also consider the x acceleration on a segment of string. If the tension is constant, then this of the order of (y')^2 (since it's a cos of a small angle) and can be safely ignored, since |y'|<<1. If you allow the tension to be variable then this is of the order of dT/dx and you can't ignore it. In fact, these variable tension modes correspond to longitudinal oscillations of the string. But that's not what this problem is about, it's about the transverse modes. If you want to do the full problem you would have to take x and y to both be functions of t and work from there. I think to really justify the approximation you would have to add a premise that the mass density of the string is sufficiently small. I think it best to keep a simple problem simple.