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Chapter 6: Problem 2

Use Lagrange's equations to derive the equation of motion of the simple pendulum.

Short Answer

Expert verified
Based on the provided solution, the second-order differential equation that describes the motion of the simple pendulum is: \ddot{θ} + \frac{g}{L} sin(θ) = 0 where θ is the angle between the rod and the vertical axis, g is the acceleration due to gravity, and L is the length of the pendulum.

Step by step solution

01

Define the System and Establish the Lagrangian Function

First, let's define the simple pendulum system. The simple pendulum consists of a mass m at the end of a massless, rigid rod of length L, fixed at one end and free to swing at the other. The angle θ between the rod and the vertical axis determines the position of the pendulum. The kinetic energy T and potential energy V of the simple pendulum are: T = 1/2 m(L \dot{θ})^2 V = -mgL cos(θ) Now, we will establish the Lagrangian function L for the simple pendulum, which is defined as the difference between the kinetic and potential energies of the system: L = T - V = 1/2 m(L \dot{θ})^2 -(-mgL cos(θ))
02

Calculate the Partial Derivatives in Lagrange's Equations

To proceed, we will calculate the required partial derivatives that appear in Lagrange's equations. These are the partial derivatives of the Lagrangian function L with respect to θ and its time derivative, \dot{θ}: 1. \frac{∂L}{∂θ} = ∂(-mgL cos(θ))/∂θ = mgL sin(θ) 2. \frac{∂L}{∂\dot{θ}} = ∂(1/2 m(L \dot{θ})^2 )/∂\dot{θ} = mL^2 \dot{θ} 3. \frac{d}{dt}(\frac{∂L}{∂\dot{θ}}) = \frac{d}{dt}(mL^2 \dot{θ}) = mL^2 \ddot{θ}
03

Apply Lagrange's Equations to Obtain the Equation of Motion

Finally, we will apply Lagrange's equations to derive the equation of motion for the simple pendulum. The general form of Lagrange's equations is: \frac{d}{dt}(\frac{∂L}{∂\dot{θ}}) - \frac{∂L}{∂θ} = 0 Plugging in the partial derivatives calculated in Step 2, we get: mL^2 \ddot{θ} - mgL sin(θ) = 0 Now, we can rearrange to find the equation of motion for the simple pendulum: \ddot{θ} + \frac{g}{L} sin(θ) = 0 This is the desired second-order differential equation describing the motion of the simple pendulum.

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Most popular questions from this chapter

Three identical rigid disks, each of mass \(m\) and radius \(R\), are attached at their centers to an elastic shaft of area polar moment of inertia \(J\) and shear modulus \(G\). The ends of the rod are embedded in rigid supports as shown. The spans between the disks and between the disks and the supports are each of length \(L\). Derive the equations of angular motion for the system if the disks are subjected to the twisting moments \(M_{1}, M_{2}\) and \(M_{3}\), respectively.

A square raft of mass \(m\) and side \(L\) sits in water of specific gravity \(\gamma_{w}\). A uniform vertical line force of intensity \(P\) acts downward at a distance \(a\) left of center of the span. (a) Use Lagrange's equations to derive the 2-D equations of motion of the raft. (b) Check your answers using Newton's Laws of Motion.

A rigid rod of length \(L\), and mass \(m_{a}\) is connected to a rigid base of mass \(m_{b}\) through a torsional spring of stiffness \(k_{T}\) as shown. The base sits on an elastic support of stiffness \(k\) as indicated. Derive the equations of motion of the system using Lagrange's equations.

Two identical bodies of mass \(m\) are connected by a spring of stiffness \(k\) and constrained to move in rectilinear motion as shown. Derive the equations of motion for the system.

Consider an aircraft traveling at constant altitude and speed as it undergoes tight periodic rolling motion of the fuselage. Let the wings be modeled as equivalent rigid bodies with torsional springs of stiffness \(k_{T}\) at the fuselage wall. In addition, let each wing possess moment of inertia \(I_{c}\) about its respective connection point and let the fuselage of radius \(R\) have moment of inertia \(I_{o}\) about its axis. Derive the equations of rolling motion for the aircraft.
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