A rigid rod of length pivots about one endpoint in a vertical plane

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A rigid rod of length L=2 mL=2\text{ m} pivots about one endpoint in a vertical plane. When the rod has rotated through a small angle θ\theta (measured in radians from the horizontal), the vertical displacement of its free endpoint is y(θ)=L(1−cos⁡θ).y(\theta)=L\bigl(1-\cos \theta\bigr). A sensor calibration requires the limit of the ratio R(θ)=y(θ)θ sin⁡θR(\theta)=\frac{y(\theta)}{\theta\,\sin \theta} as θ→0\theta\to 0. (a) Show that R(θ)R(\theta) is undefined at θ=0\theta=0 but is expressible near θ=0\theta=0 as a quotient of two functions that both approach 00 as θ→0\theta\to 0. (b) Determine lim⁡θ→0R(θ)\displaystyle\lim_{\theta\to 0}R(\theta) with complete justification. (c) Interpret the numerical value obtained in part (b) in terms of the small–angle relationship between the rod’s vertical displacement and the product θsin⁡θ\theta\sin \theta. Explain how this result informs the engineer’s calibration of the displacement sensor for very small rotations.

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