Physics

Physics

Problem

A torsion pendulum consists of a rigid bar that can rotate about its center. Four identical small masses can be slid along the bar and locked into place at various distances from the center. The restoring torque provided by the torsion wire can be considered negligible for small angular displacements. Students measure the static torque required to hold the bar at different angular positions θ\theta for a particular arrangement of masses. Their results are summarized verbally as follows: • Between θ=−30∘\theta=-30^{\circ} and θ=+30∘\theta=+30^{\circ}, the measured torque varies approximately linearly with θ\theta and crosses zero at θ=+8∘\theta=+8^{\circ}. • For θ>+30∘\theta>+30^{\circ}, the torque becomes increasingly positive at a faster rate. • For θ<−30∘\theta<-30^{\circ}, the torque becomes increasingly negative at a faster rate. Positive torque corresponds to counter-clockwise rotation. (a) On labeled axes, sketch the qualitative shape of the torque–angle graph implied by the description above. Indicate all intercepts and notable changes in slope. (b) Using the graph you sketched, identify every angle within −30∘≤θ≤+30∘-30^{\circ}\le\theta\le+30^{\circ} at which the bar could be in static rotational equilibrium. Explain your reasoning. (c) Classify each equilibrium you identified in part (b) as stable or unstable. Justify each classification by referring to the behavior of the torque for small displacements from the equilibrium point. (d) Suppose the students symmetrically reposition the masses farther from the center, doubling each mass’s moment arm. Predict qualitatively how the slope of the linear region of the torque–angle graph would change, and explain the physical basis for your prediction.
Intuitive explanation

The key insight is recognizing that the restoring torque comes from gravity on the off-center masses, not the torsion wire. When the bar tilts, each mass shifts its horizontal position relative to the pivot, creating a gravitational torque that depends on both the mass distance from center and the tilt angle. The linear region tells you the small-angle approximation holds there, while the nonlinear behavior at larger angles reveals how the gravitational lever arm changes—this is where sin⁡θ\sin\theta effects become important rather than just θ\theta itself.

Intuitive explanation

The key insight is recognizing that the restoring torque comes from gravity on the off-center masses, not the torsion wire. When the bar tilts, each mass shifts its horizontal position relative to the pivot, creating a gravitational torque that depends on both the mass distance from center and the tilt angle. The linear region tells you the small-angle approximation holds there, while the nonlinear behavior at larger angles reveals how the gravitational lever arm changes—this is where sin⁡θ\sin\theta effects become important rather than just θ\theta itself.

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