Question:easy

Consider a single degree of freedom system with viscous damping excited by a harmonic force. At resonance, the phase angle of the displacement with respect to the exciting force in degrees is

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Phase angle (\(\phi\)) transitions relative to frequency ratio (\(r = \frac{\omega}{\omega_n}\)): - Low frequencies (\(r \ll 1\)): \(\phi \to 0^\circ\) (Response is in phase with force). - At Resonance (\(r = 1\)): \(\phi = 90^\circ\) exactly, regardless of the value of damping ratio \(\zeta\) (provided \(\zeta \gt 0\)). - High frequencies (\(r \gg 1\)): \(\phi \to 180^\circ\) (Response is out of phase).
Updated On: Jul 4, 2026
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The Correct Option is C

Solution and Explanation

Picture the mass moving up and down under the harmonic force \(F_0 \sin(\omega t)\). The displacement response can be written as a wave that lags the force by some angle \(\phi\), where \(\tan \phi = \dfrac{2 \zeta r}{1 - r^2}\) and \(r = \omega/\omega_n\) is the frequency ratio. Resonance happens exactly when the driving frequency matches the natural frequency, so \(r = 1\). Putting \(r = 1\) into the formula makes the denominator \(1 - r^2\) go to zero, so the tangent of the phase angle shoots up to infinity. The only angle in this range where \(\tan \phi\) is infinite is \(90^\circ\). This also matches the physical picture: at resonance, the applied force is used up entirely in beating the damping force, and since the damping force is always in phase with velocity, the displacement ends up a quarter cycle, that is \(90^\circ\), behind the force. Hence the answer is \(90^\circ\), option (3).
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