GATE 2021 AE – Question 30
Two cantilever beams made of the same homogeneous material have identical cross sections and lengths L and 2L. The ratio of the first natural frequency of the shorter beam to the longer beam is (nearest integer)
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Correct answer: 4
Explanation
The natural frequency of a cantilever beam is
$$\omega_n=\beta_n^2\sqrt{\frac{EI}{\rho A\,L^4}},$$
where $\beta_nL$ is a constant for the mode. For the same material and cross section ($E$, $I$, $\rho$, $A$ fixed), $\omega_n\propto\dfrac1{L^2}$.
**Ratio** of the first natural frequency of the shorter beam (length $L$) to that of the longer beam (length $2L$):
$$\frac{\omega_L}{\omega_{2L}}=\left(\frac{2L}{L}\right)^2=\mathbf{4}.$$