Small-angle approximation

Small-angle approximation refers to a mathematical technique used to simplify calculations involving angles that are very small. It is a method of approximating the value of trigonometric functions such as sine, cosine, and tangent when the angle is small enough that the difference between the actual value and the approximation is negligible. This technique is commonly used in physics, engineering, and other fields where precise calculations are required.From: Oscillations and Waves [2019]

Second-Order Ordinary Differential Equations

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Published in Brian Vick, Applied Engineering Mathematics, 2020

Brian Vick

Often, the small angle approximation, sin(θ)≅θ, is made, reducing the equation of motion to a linear system. How about the large angle behavior?

The kinematics and static equilibria of a Slinky

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Published in International Journal of Mathematical Education in Science and Technology, 2022

Peter Cumber

The spring constants kr and kpt are calibrated to reproduce the length of the Slinkies in a vertical orientation. For example, the plastic Slinky has a self-weight vertical length of 0.535 m. In Figure 5(a) the plastic Slinky shape is calculated using the torsional spring model satisfying the energy equation (8) and the small angle approximation (20). To make it possible to see the two predictions of the Slinky in Figure 5(a) an origin shift of 0.2 m is applied to the small angle approximation Slinky. The small angle approximation Slinky predicts the correct length but the orientation is incorrect. This is understandable as for coils near the bottom of the Slinky, the small angle approximation is an accurate assumption. At the fixed end near the origin the torsion angles are large due to the self-weight and the small angle approximation is no longer valid. In Figure 5(b) a mini plastic Slinky with 15 coils in a vertical orientation is shown. In this case the small angle approximation works very well.

Small-angle approximation

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Second-Order Ordinary Differential Equations

The kinematics and static equilibria of a Slinky