Error Propagation Homework: Find T's Error

[bigevil]

Homework Statement

For my lab work, I have created a theoretical model that goes something like:

$$T = \sqrt{\frac{ks^2}{x \sin \theta \cos^2\theta}}$$

where k is a constant, and the variables to be differentiated are x, theta and s. How do I find the error of T? I can find the errors of x and s (dx/x and ds/s) from experiment. And of course

$$d\theta / \theta$$ can also be found from the instrument.

The Attempt at a Solution

Differentiating the term for theta,

$$\frac{dF}{d\theta} = -\frac{1}{2} \sqrt{\frac{1}{\sin\theta \cos^2 \theta}} \cos\theta (1 - 3\sin^2\theta)$$

It's easier to differentiate the other two. Anyway, I have:

$$\frac{dT}{T} = \frac{ds}{s} + \frac{1}{2}\frac{dx}{x} + \frac{(3\sin^2\theta - 1)\cos\theta}{2}d\theta$$

How can I express the last term in a $$d\theta / \theta$$ form? The only thing I can think of at the moment is using a small angle approximation, but I don't know how to justify that. And also, clearly, if I could use that, the small angle approximation for cosine has a square term.

 

[RoyalCat]

For a function of multiple variables, $$T=f(x, s, \theta)$$, the error in the function, $$\Delta T=\sqrt{(\frac{\partial T}{\partial x}\cdot \Delta x)^2+(\frac{\partial T}{\partial s}\cdot \Delta s)^2+(\frac{\partial T}{\partial \theta}\cdot \Delta \theta)^2}$$

I suggest that you stop using the lowercase $$d$$ for the errors, because you're bound to get it mixed up with the derivatives you're taking.

 

[kerimek]

To find the error of T, we need to use the error propagation formula:

δT = √( (∂T/∂x)² * δx² + (∂T/∂θ)² * δθ² + (∂T/∂s)² * δs² )

where δx, δθ, and δs are the errors in x, θ, and s respectively.

Using the chain rule, we can find the partial derivatives of T with respect to each variable:

∂T/∂x = (1/2) * √(k/s² * sinθ * cos²θ) * (-1/x²) = -(1/2) * T/x

∂T/∂θ = (1/2) * √(k/s² * cos²θ / sin³θ) * (1 - 3sin²θ) = (1/2) * T * (1 - 3sin²θ) / (sinθ * cos²θ)

∂T/∂s = (1/2) * √(k/s² * sinθ * cos²θ) * (-2s/s³) = -(1/s) * T

Substituting these values into the error propagation formula, we get:

δT = √( (-1/2) * T/x * δx² + (1/2) * T * (1 - 3sin²θ) / (sinθ * cos²θ) * δθ² + (-1/s) * T * δs² )

We can express the last term in the form of dθ/θ by using the small angle approximation for sine and cosine. This approximation states that for small angles, sinθ ≈ θ and cosθ ≈ 1. Therefore, we can rewrite the last term as:

(-1/s) * T * δs² = (-1/s) * T * (dθ/θ * θ)² ≈ (-1/s) * T * (dθ)²

Substituting this into the error propagation formula, we get:

δT = √( (-1/2) * T/x * δx² + (1/2) * T * (1 - 3θ²) * δθ² + (-1/s) * T * δs² )

This