The misalignment of the amplitudes between similarly behaving pendulums is called Group of answer choices Steady-state motion Wavelength Resonance Phase shift
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(a) A force $F$ is applied at point $A$ of a pendulum as shown. At what angle $\theta(\ll 1 \mathrm{rad})$ is the new equilibrium position? What force $F^{\prime}$, applied at $m$, would produce the same result? (a) (b) Two identical pendulums consisting of equal masses mounted on rigid, weightless rods, are arranged as shown. A light spring (unstretched when both rods are vertical, and placed as shown) provides the coupling. (b) Write down the differential equations of motion for smallamplitude oscillations in terms of $\theta_{1}$ and $\theta_{2}$. (Neglect damping.) (c) Describe the motion of the pendulums in each of the normal modes. (d) Calculate the frequencies of the normal modes of the system. [Hint: The symmetry of the system can be exploited to good advantage, particularly in parts (c) and (d), as long as the answers obtained this way are checked in the equations.
In the coupled pendulums of Figure $4.3$ let us write the modulated frequency $\omega_{m}=\left(\omega_{2}-\omega_{1}\right) / 2$ and the average frequency $\omega_{a}=\left(\omega_{2}+\omega_{1}\right) / 2$ and assume that the spring is so weak that it stores a negligible amount of energy. Let the modulated amplitude $$ 2 a \cos \omega_{m} t \quad \text { or } \quad 2 a \sin \omega_{m} t $$ be constant over one cycle at the average frequency $\omega_{a}$ to show that the energies of the masses may be written $$ E_{x}=2 m a^{2} \omega_{a}^{2} \cos ^{2} \omega_{m} t $$ and $$ E_{y}=2 m a^{2} \omega_{a}^{2} \sin ^{2} \omega_{m} t $$ Show that the total energy $E$ remains constant and that the energy difference at any time is $$ E_{x}-E_{y}=E \cos \left(\omega_{2}-\omega_{1}\right) t $$ Prove that $$ E_{x}=\frac{E}{2}\left[1+\cos \left(\omega_{2}-\omega_{1}\right) t\right] $$ and $$ E_{y}=\frac{E}{2}\left[1-\cos \left(\omega_{2}-\omega_{1}\right) t\right] $$ to show that the constant total energy is completely exchanged between the two pendulums at the beat frequency $\left(\omega_{2}-\omega_{1}\right)$.
57. Amplitude Ratio In Touchstone Example $16-4$, what is the ratio of the amplitude of the damped oscillations to the initial amplitude when 20 full oscillations have elapsed?
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