Understanding Approximations in Angular Motion Equations

[Joshuarr]

Homework Statement

It's attached. The problem and solution are given.

Homework Equations

The Attempt at a Solution

I circled a part of the image in red. Is this substitution supposed to be an approximation?

I was thinking it was because one is referring to angular motion, so we're tracing out an arc, and the x is vertically linear, but I guess since it's such a small angle that it's not much different. If this is an approximation, is this the same approximation where we use sin θ ≈ θ, for small θ?

 

[voko]

I was thinking it was because one is referring to angular motion, so we're tracing out an arc, and the x is vertically linear, but I guess since it's such a small angle that it's not much different. If this is an approximation, is this the same approximation where we use sin θ ≈ θ, for small θ?

The equation they used relates the tangential acceleration and angular acceleration, and this equation is exact in the frame of reference co-rotating with the rod. So no approximation here. A lot of approximation appeared earlier, in the passage that starts with "Now let us mentally erect a vertical x axis...". It is obvious that due to the rotation of the rod, the spring also rotates, so the force it creates is not just proportional to x. Moreover, the direction of the force is not perpendicular to the rod. Another approximation happens after it, in the passage that says "Equation 15-36 is, in fact, of the same form..." The form may be similar, but a is not x'', because they are in different frames of reference. You can work these details out and then you will see what those approximation really were. Yes, they are of the kind that you mentioned.