(6) Pendulums with spring: Derive the 2 equations of motion for the 2 pendulum system shown: $L_1$ $\theta_1(t)$ $L_2$ $m_1$ $L_2$ $L_1$ $\theta_2(t)$ $m_2$
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The position of mass $m_1$ is given by $(x_1, y_1)$ and the position of mass $m_2$ is given by $(x_2, y_2)$. We have: $x_1 = L_1 \sin(\theta_1)$ $y_1 = -L_1 \cos(\theta_1)$ $x_2 = L_1 \sin(\theta_2)$ $y_2 = -L_1 \cos(\theta_2)$ Show more…
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4.6 Consider a pair of pendulums, one twice as long as the other, coupled to each other by a spring. Each pendulum consists of a massless stiff rod with a mass m at its end. The spring is attached halfway along the length of the longer pendulum and to the end of the other pendulum as shown in the figure below. Assume that ̘̘1 = ̘̘2 = 0 and that the spring is unstretched at equilibrium. (a) Write down the coupled equations of motion for the two pendulums, using the angles ̘̘1 and ̘̘2 as the dynamical variables and assuming that ̘̘1 << 1 and ̘̘2 << 1. Write the coupled equations of motion in angular form using torque ̘ = rF, that is, in the form ̘ = I̘, where I is the moment of inertia. Show that the equations of motion can be written in matrix form m ̘ = -k̘.
Adi S.
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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.
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