A horizontal turntable can rotate about a fixed vertical axis

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems

Problem

A horizontal turntable can rotate about a fixed vertical axis. While the turntable is at rest, a variable net torque is applied. Over the interval 0≤θ≤4π rad0 \le \theta \le 4\pi\,\text{rad} the torque depends on angular position according to the following description: • From θ=0\theta = 0 to θ=2π rad\theta = 2\pi\,\text{rad} the torque increases linearly from 0 N⋅m0\,\text{N}\cdot\text{m} to 0.60 N⋅m0.60\,\text{N}\cdot\text{m}. • From θ=2π rad\theta = 2\pi\,\text{rad} to θ=4π rad\theta = 4\pi\,\text{rad} the torque decreases linearly back to 0 N⋅m0\,\text{N}\cdot\text{m}. The turntable has moment of inertia 0.12 kg⋅m20.12\,\text{kg}\cdot\text{m}^2 and starts from rest. Ignore air resistance and bearing friction unless otherwise stated. (a) On axes of torque (vertical) versus angular position (horizontal), sketch the qualitative shape of the torque function described. Label key values on both axes. (b) Calculate the net mechanical work done on the turntable by the applied torque as it turns through 4π rad4\pi\,\text{rad}. (c) Determine the angular speed of the turntable after it has turned through 4π rad4\pi\,\text{rad}. (d) Suppose a constant opposing friction torque of 0.050 N⋅m0.050\,\text{N}\cdot\text{m} acts on the axis during the same interval. Recalculate the angular speed at θ=4π rad\theta = 4\pi\,\text{rad} incorporating this additional torque.

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