Question

In Rayleigh's method, which of the following energy at the mean position is equal to the maximum potential energy (or strain energy) at the extreme position?

• A. Maximum kinetic energy
• B. Minimum kinetic energy
• C. Maximum potential energy
• D. Minimum potential energy

Hint:

Rayleigh's Method summary formula: \[ \text{K.E.}_{\max} = \text{P.E.}_{\max} \] For a lump mass system with displacement \(x = X\sin(\omega_n t)\): \[ \frac{1}{2} m (\omega_n X)^2 = \frac{1}{2} k X^2 \quad \Rightarrow \quad \omega_n = \sqrt{\frac{k}{m}} \]

Answer 1

The Correct Option is A

Take a mass vibrating in simple harmonic motion, \(x = X \sin(\omega t)\), with no damping so the total mechanical energy stays constant throughout the cycle. At the mean position the displacement is zero, so all the spring's potential energy has already been given back and the potential energy there is zero, while the velocity is at its largest value \(X\omega\), so the kinetic energy is at its peak. At the two extreme ends of travel the mass is momentarily at rest, so its kinetic energy is zero, and all the energy the system has is stored as potential (strain) energy. Because no energy is lost anywhere in this ideal system, the energy at the mean position must equal the energy at the extreme position, which means the maximum kinetic energy equals the maximum potential energy. This equality is exactly what Rayleigh's method uses to estimate the natural frequency, so the answer is maximum kinetic energy, option (1).