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A simple pendulum has a length L and a b...

A simple pendulum has a length L and a bob of mass m. The bob is oscillating with amplitude A. Show that the maximum tension (T) in the string is (for small angular displacement).
`T_("max") =mg [1+(A/L)^(2)]`.

Text Solution

Verified by Experts

Here, Mass of bob =m, Amplitude of oscillation = A, Length of pendulum = L, `/_BAC = theta`

Tension `T= mg cos theta +(mv^2)/(L)" "[(mv^(2))/(L) = " centripetal force "]`
If `cos theta =1`
`v= v_("max") " and " T= T_("max")`
`T_("max") = mg +(mv_("max")^(2))/(L)" ""..........."(1)`
`=mg +(mA^(2)omega^(2))/(L)" but "omega^(2) = (g)/(L)`
`T_("max") = mg+ (mA^(2) g)/(L xx L)" "".........."(2)`
`therefore T_("max") = mg +(mgA^(2))/(L^2)`
`therefore T_("max") = mg[1+(A/L)^(2)]`.
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