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How does the length of a cantilever affect the period of osscillation?

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year8 | (Level 1) Honors

Posted February 26, 2013 at 8:56 AM via web

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How does the length of a cantilever affect the period of osscillation?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted May 20, 2013 at 5:22 AM (Answer #1)

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You need to use the following equation that gives the theoretical formula of oscillation of cantilever, such that:

`T = 2pi*sqrt((m*L^3)/(3EI))`

`L` represents the length of cantilever

`m` represents the mass that affects the cantilever

`EI` represents the flexural rigidity

You should notice that L represents the length of cantilever and it is directly proportional to period of oscillation T, hence, the more the length of cantilever is, the longer period of oscillation of cantilever is.

Hence, investigating two periods `T_1, T_2` , at different lengths `L_1,L_2` , yields:

`L_1 < L_2`

`L_1 = 1m, L_2 = 2m`

`T_1 = 2pi*sqrt((m*L_1^3)/(3EI)) => T_1 = 2pi*sqrt((m)/(3EI))`

`T_2 = 2pi*sqrt((m*2^3)/(3EI)) => T_2 = 4sqrt2pi*sqrt((m)/(3EI))`

Comapring `T_1` and `T_2` yields:

`2pi*sqrt((m)/(3EI)) ~ 4sqrt2pi*sqrt((m)/(3EI)) => T_1 = 1 < T_2 = 2sqrt 2`

Hence, evaluating the periods `T_1, T_2` for` L_1<L_2` yields that `T_1<T_2` , thus, the period of oscillation of cantilever T is directly proportional with the length L.

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