Physics

Physics

Problem

A 0.50 m long bar is rotated horizontally about one end while a string pulls on the free end with a constant 12 N force. As the bar extends outward, the angle between the string and the bar decreases uniformly from 90° at r = 0.10 m to 60° at r = 0.50 m. Predict whether a graph of torque magnitude versus radial distance over this interval would be linear, concave upward, or concave downward, and explain the physical reason for the shape you identify.
Intuitive explanation

Torque depends on both the lever arm (perpendicular distance from the pivot) and the radial distance where force is applied—here, τ=Frsin⁡θ\tau = Fr\sin\theta, where θ\theta decreases as rr increases. While the rr factor grows linearly, the sin⁡θ\sin\theta factor shrinks, creating competing effects. The key is recognizing that sin⁡θ\sin\theta decreases nonlinearly (slowly at first, then faster), so the product rsin⁡θr\sin\theta initially grows faster than it would if θ\theta stayed constant, then flattens out—making the curve concave downward. This reveals how geometry can overwhelm the simple linear scaling you might expect.

Intuitive explanation

Torque depends on both the lever arm (perpendicular distance from the pivot) and the radial distance where force is applied—here, τ=Frsin⁡θ\tau = Fr\sin\theta, where θ\theta decreases as rr increases. While the rr factor grows linearly, the sin⁡θ\sin\theta factor shrinks, creating competing effects. The key is recognizing that sin⁡θ\sin\theta decreases nonlinearly (slowly at first, then faster), so the product rsin⁡θr\sin\theta initially grows faster than it would if θ\theta stayed constant, then flattens out—making the curve concave downward. This reveals how geometry can overwhelm the simple linear scaling you might expect.

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