I Why is my derivation of the catenary wrong?

AI Thread Summary
The discussion centers on the differentiation of the tangent function, specifically addressing a common mistake in its approximation. The tangent approximation used, tan(theta + dtheta) ~ tan(theta) + dtheta, is identified as incorrect. The proper differentiation of tan(theta) should involve the quotient rule, leading to a more accurate expression. Participants agree on the need for careful application of differentiation rules in this context. The emphasis is on understanding the correct mathematical approach to avoid errors in solving differential equations.
phantomvommand
Messages
287
Reaction score
39
TL;DR Summary
I have "derived" a differential equation for the catenary, and have attached my working. It looks slightly different from the correct expression, which can be found here: https://www.math24.net/equation-catenary

Please do tell me where I made a mistake. Thank you!
Important note: I only derived the differential equation, I did not solve it.

WhatsApp Image 2021-03-04 at 1.22.37 AM.jpeg

What I think caused the mistake:
- the tangent approximation (tan(theta+dtheta) ~ tan theta + d theta
 
Physics news on Phys.org
Hi,

phantomvommand said:
What I think

That's what I think too :wink:
The proper way to differentiate ##\tan\theta## is not ##{d\tan\theta\over d\theta} = 1 ## but $${d\tan\theta\over d\theta} = {d\over d\theta}\Biggl ( {\sin\theta\over\cos\theta}\Biggr ) =\ ... $$
​
##\ ##​
 

Thread 'The Effect of Linear and Rotational Motion on Measured Weight'

Consider an extremely long and perfectly calibrated scale. A car with a mass of 1000 kg is placed on it, and the scale registers this weight accurately. Now, suppose the car begins to move, reaching very high speeds. Neglecting air resistance and rolling friction, if the car attains, for example, a velocity of 500 km/h, will the scale still indicate a weight corresponding to 1000 kg, or will the measured value decrease as a result of the motion? In a second scenario, imagine a person with a...
View full post »

Thread 'Coulomb gauge implies instantaneous radial electric field?'

Scalar and vector potentials in Coulomb gauge Assume Coulomb gauge so that $$\nabla \cdot \mathbf{A}=0.\tag{1}$$ The scalar potential ##\phi## is described by Poisson's equation $$\nabla^2 \phi = -\frac{\rho}{\varepsilon_0}\tag{2}$$ which has the instantaneous general solution given by $$\phi(\mathbf{r},t)=\frac{1}{4\pi\varepsilon_0}\int \frac{\rho(\mathbf{r}',t)}{|\mathbf{r}-\mathbf{r}'|}d^3r'.\tag{3}$$ In Coulomb gauge the vector potential ##\mathbf{A}## is given by...
View full post »

Thread 'Simple thought experiment with Stefan-Boltzmann law: energy'

Dear all, in an encounter of an infamous claim by Gerlich and Tscheuschner that the Greenhouse effect is inconsistent with the 2nd law of thermodynamics I came to a simple thought experiment which I wanted to share with you to check my understanding and brush up my knowledge. The thought experiment I tried to calculate through is as follows. I have a sphere (1) with radius ##r##, acting like a black body at a temperature of exactly ##T_1 = 500 K##. With Stefan-Boltzmann you can calculate...
View full post »

Similar threads

I What is the Unsolvable Catenary Problem?
Replies
17
Views
4K
B Sign conventions in torque and non-uniform circular motion
Replies
1
Views
1K
I Transformations for isotropy in terms of math
Replies
1
Views
1K
I Why isn't the Lagrangian invariant under ##\theta## rotations?
Replies
5
Views
2K
I Mathematical disagreement with inverse square law?
Replies
33
Views
2K
B Problems with modelling the vertical motion of a dust particle
Replies
15
Views
837
I What assumption is incorrect in the derivation of the catenary curve equation?
Replies
15
Views
3K
I Question on derivation of a property of Poisson brackets
Replies
0
Views
1K
Calculation of Thomas Precession in Rindler's relativity book
Replies
6
Views
1K
I My textbook is deriving wave speed on a string under tension
Replies
19
Views
2K

Hot Threads

Recent Insights

Back
Top