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A stone is tied to a string of length ℓ and is whirled in a vertical circle with the other end of the string as the center. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of the change in velocity as it reaches a position where the string is horizontal (g being acceleration due to gravity) is

1. √2gℓ
2. √2(u² − gℓ)
3. u − √u² − 2gℓ
4. √u² − gℓ

Correct answer: √2(u² − gℓ)

Solution

The change in velocity is determined by the difference in the speed of the stone at the lowest point and when the string is horizontal. Using energy conservation, the speed at the horizontal position is √(u² - 2gℓ). The magnitude of the change in velocity is then √2(u² − gℓ).

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