PQ ≈ Arc Length: Explained

[autodidude]

Could someone please explain why PQ in the diagram below is rΔθ? Isn't rΔθ arc length?

The best reason I can think of is that it's only an approximation for when the angle is very small, so PQ≈arclength=rΔθ. Not 100% sure though.

http://imageshack.us/scaled/landing/199/feynmanangle.jpg

The diagram is from the first volume of the Feynman lectures in 18-3, in the section where he talks about rotation of rigid bodies.

 

[mfb]

It is an approximation for small $\Delta \theta$, right. I would expect that it is used in a differential somewhere, where the approximation gets exact.

 

[autodidude]

Thanks mfb

 

[rcgldr]

Start with the actual distance:

PQ = sqrt(Δr² + (r sin(Δθ))²)

If this is circular motion, then Δr = 0, and as Δθ -> 0, then sin(Δθ) -> Δθ, and you end up with lim Δθ -> 0 of sqrt((r sin(Δθ))²) -> sqrt((r Δθ)²) -> r Δθ.

If r is some function of θ, then as long as Δr approaches zero more rapidly than r sin(Δθ), then lim Δθ -> 0 of f(Δr, r sin(Δθ)) -> f(0, r sin(Δθ)).

 

[CellCoree]

Yes, you are correct. In this diagram, PQ is an approximation of the arc length, which is equal to rΔθ. This approximation holds true when the angle Δθ is very small. As the angle increases, the approximation becomes less accurate. This concept is known as the small angle approximation and is commonly used in mathematics and physics to simplify calculations. In this case, it allows us to approximate the arc length using a simpler formula, rΔθ, which is easier to work with. However, it is important to keep in mind that this is only an approximation and may not be accurate for larger angles.