A student plots torque versus angle for a wrench as the angle increases from

MATHEMATICS · Single correct · A Level · Mathematics · Trigonometry · Trigonometry

Problem

A student plots torque τ\tau versus angle θ\theta for a wrench as the angle increases from 0 rad0\,\text{rad}. The graph is nearly linear for small angles before curving upward toward a maximum. Which of the following best explains why the initial portion of the graph is almost a straight line? (a) Because as the wrench first begins to turn, the person typically pushes harder, and with τ=rF\tau = rF this makes τ\tau grow almost proportionally with θ\theta. (b) Because for small angles sin⁡θ≈θ\sin\theta \approx \theta, so τ=rFsin⁡θ\tau = rF\sin\theta increases nearly linearly with θ\theta. (c) Because at small angles cos⁡θ≈1\cos\theta \approx 1, so the lever arm is essentially constant and any rise in force makes τ\tau appear to vary linearly with θ\theta. (d) Because torque equals IαI\alpha, and for small rotations the angular acceleration is nearly constant, resulting in τ\tau increasing roughly linearly with the angle turned.

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