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Question
The period of a conical pendulum in terms of its length (l), semi-vertical angle (θ) and acceleration due to gravity (g) is ______.
Options
`1/(2 pi) sqrt((l cos theta)/g)`
`1/(2 pi)sqrt((l sin theta)/g)`
`4 pi sqrt((l cos theta)/(4 g))`
`4 pi sqrt((l tan theta)/g)`
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Solution
The period of a conical pendulum in terms of its length (l), semi-vertical angle (θ) and acceleration due to gravity (g) is `bbunderline(4 pi sqrt((l cos theta)/(4 g)))`.
Explanation:
For a conical pendulum, the vertical component of tension balances weight:
T cos θ = mg
= `4 pi sqrt ((l cos theta)/g)`
= `4 pi (1/2) sqrt((l cos theta)/g)`
= `2 pi sqrt((l cos theta)/g)`
Using centripetal force and angular motion relations, we get the time period:
T = `2 pi sqrt((l cos theta)/g)`
∴ The expression `4 pi sqrt ((l cos theta)/g)` is mathematically identical to the standard period formula for a conical pendulum.
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