Vertical circular motion – at which position is string tension maximum? A stone is whirled in a vertical circle using a light, inextensible string. Considering both weight and centripetal requirements, at which position of the path is the tension in the string greatest?

Difficulty: Easy

when the string is horizontal
when the stone is at the highest position
when the stone is at the lowest position
at all positions the tension is the same
when the stone is at a 45° position

Correct Answer: when the stone is at the lowest position

Explanation:

Introduction / Context:
In vertical circular motion, the string must supply the inward (centripetal) force while also countering or being aided by the weight of the body. Understanding how tension varies around the path is essential for safe design of rotating machinery, amusement rides, and cable-driven systems.

Given Data / Assumptions:

Light, inextensible string and a small stone treated as a particle.
Motion is in a vertical plane with speed possibly varying due to gravity.
Neglect air resistance and any stiffness effects in the string.

Concept / Approach:

Tension is not constant in vertical circular motion. At any point, the centripetal requirement is m * v^2 / r toward the center. Weight mg acts downward. The string tension T must combine correctly with weight to provide the needed inward acceleration. Positions of special interest are the bottom and the top of the circle because the weight reverses its contribution relative to the required inward direction.

Step-by-Step Solution:

At the lowest point: inward direction is upward, so T_bottom − mg = m * v_bottom^2 / r → T_bottom = mg + m * v_bottom^2 / r.At the highest point: inward direction is downward, so T_top + mg = m * v_top^2 / r → T_top = m * v_top^2 / r − mg.Since v_bottom ≥ v_top for the same energy (gravity speeds the stone on the way down), and mg adds at the bottom but subtracts at the top, T_bottom is the largest.At horizontal positions the component of weight toward the center is mg * sinθ and does not exceed mg; tension there is between top and bottom values.

Verification / Alternative check:

Energy conservation between top and bottom shows v_bottom^2 = v_top^2 + 4 g r. Substituting into the expressions above confirms T_bottom − T_top = 2 mg + 4 m g = 6 m g minus terms depending on speeds, ensuring T_bottom > T_top for feasible motion.

Why Other Options Are Wrong:

Horizontal (a): tension is intermediate. Highest point (b): tension is least and may even approach zero at minimum-speed conditions. Same everywhere (d): false in vertical motion. 45° (e): not the extremum.

Common Pitfalls:

Assuming constant speed as in horizontal whirling; forgetting the weight reverses its role at top vs. bottom; mixing up inward (centripetal) direction with tension direction.

Final Answer:

when the stone is at the lowest position

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Vertical circular motion – at which position is string tension maximum? A stone is whirled in a vertical circle using a light, inextensible string. Considering both weight and centripetal requirements, at which position of the path is the tension in the string greatest?

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