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(a) A thin uniform rod of length 2 b is suspended by two vertical light strings, both of fixed length l, fastened to the ceiling. Assuming only small displacements from equilibrium, find the Lagrangian of the system and the normal frequencies. Find and describe the normal modes. [Hint: A possible choice of generalized coordinates would be x, the longitudinal displacement of the rod, and y1 and y2, the sideways displacements of the rod's two ends. You'll need to find how high the two ends are above their equilibrium height and what angle the rod has turned through.]

Short Answer

Expert verified
Lagrangian involves kinetic and potential energies. Solve Euler-Lagrange equations for normal frequencies and modes.

Step by step solution

01

Understand the Problem

The problem involves a thin uniform rod suspended by two strings. We need to find the Lagrangian and the normal frequencies and modes. The hint suggests using three generalized coordinates: x (longitudinal displacement), y1 and y2 (sideways displacements).
02

Set Up Generalized Coordinates

Let x be the longitudinal displacement of the entire rod. The sideways movements of the two ends are noted as y1 and y2. The rod turns slightly, introducing a small angle θ.
03

Determine the Potential Energy

The potential energy for the system can be expressed in terms of the heights of the two ends: V=12mg(2l(height differences of ends)), where m is the mass of the rod and g is the gravitational constant.
04

Calculate the Kinetic Energy

The kinetic energy is the sum of translational and rotational energies: T=12mx˙2+12m((l2θ˙+y1˙)2+(l2θ˙+y2˙)2).
05

Formulate the Lagrangian

The Lagrangian L is calculated by subtracting the potential energy from the kinetic energy: L=TV.
06

Derive the Equations of Motion

Using the Euler-Lagrange equation for each generalized coordinate yields a system of differential equations reflecting the system's dynamics.
07

Find the Normal Modes and Frequencies

Analyze the equations found by assuming solutions of the form: x=Aeiωt, y1=Beiωt, and y2=Ceiωt. This leads to a matrix equation Mω2=K, from which you can solve for ω to find normal frequencies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Modes
Normal modes are a fascinating concept in physics that describe the distinct patterns of motion that a system can exhibit when disturbed from its equilibrium. Each of these patterns vibrates at a specific frequency where all parts of the system move together, in sync. For the rod suspended by two strings, you can imagine normal modes as specific ways the rod sways or pivots simultaneously.

Understanding normal modes involves:
  • Recognizing that each mode represents a distinct set of motion
  • Identifying that in a given mode, all parts of the system oscillate with the same frequency
In our problem, the two strings set constraints that dictate these characteristic movements. The analysis of normal modes is crucial since any small disturbance in a system can be decomposed into these modes. It simplifies understanding how complicated systems behave as a collection of simpler oscillations. Each mode is associated with a normal frequency, which is determined from the mathematical solution of the system’s equations of motion.
Normal Frequencies
Normal frequencies correspond to the distinct rates at which each normal mode of the system vibrates. Imagine each normal mode as having its own unique soundtrack—the normal frequency would be like the tempo of that melody.

The solution to our problem shows how to find these frequencies by solving the resulting 'matrix equation' from our differential equations of motion.

Normal frequencies are important because:
  • They tell us how fast the system would naturally want to oscillate in each possible motion pattern
  • Allow for predicting almost periodic motions without resorting to damping or external forces
In the Lagrangian approach, solving for normal frequencies involves simplifying complex motions into simpler, independent differential equations. The beauty of mathematical physics is how it transforms a complicated physical setup into a straightforward calculation where these frequencies emerge naturally.
Generalized Coordinates
Generalized coordinates are a powerful tool in mechanics, allowing for an adaptable description of a system's configuration with the least amount of description necessary. Rather than being restricted to traditional Cartesian coordinates, generalized coordinates grant the flexibility to use any natural or convenient parameters to describe the system's state.

In this exercise:
  • We use three coordinates: x, y1, and y2, which are strategically chosen to make calculations straightforward
  • The x coordinate represents longitudinal movement, while y1 and y2 are the sideways displacements of the rod's ends
By utilizing these coordinates, we can express both the system's potential and kinetic energy in simplified terms. This choice simplifies the process of deriving the Lagrangian, making subsequent analyses, such as finding normal modes and frequencies, more approachable. Generalized coordinates shine in complex systems, reducing the degrees of freedom to a practical minimum, which is particularly beneficial for solving dynamical equations in Lagrangian mechanics.

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Most popular questions from this chapter

A thin rod of length 2 b and mass m is suspended by its two ends with two identical vertical springs (force constant k ) that are attached to the horizontal ceiling. Assuming that the whole system is constrained to move in just the one vertical plane, find the normal frequencies and normal modes of small oscillations. Describe and explain the normal modes. [Hint: It is crucial to make a wise choice of generalized coordinates. One possibility would be r,ϕ, and α, where r and ϕ specify the position of the rod's CM relative to an origin half way between the springs on the ceiling, and α is the angle of tilt of the rod. Be careful when writing down the potential energy.].

Consider two equal-mass carts on a horizontal, frictionless track. The carts are connected to each other by a single spring of force constant k, but are otherwise free to move freely along the track. (a) Write down the Lagrangian and find the normal frequencies of the system. Show that one of the normal frequencies is zero. (b) Find and describe the motion in the normal mode whose frequency is nonzero. (c) Do the same for the mode with zero frequency. [Hint: This one requires some thought. It isn't immediately clear what oscillations of zero frequency are. Notice that the eigenvalue equation (Kω2M)a=0 reduces to Ka=0 in this case. Consider a solution x(t)=af(t), where f(t) is an undetermined function of t and use the equation of motion, Mx¨=Kx, to show that this solution represents motion of the whole system with constant velocity. Explain why this kind of motion is possible here but not in the previous examples.]

A massless spring (force constant k1 ) is suspended from the ceiling, with a mass m1 hanging from its lower end. A second spring (force constant k2 ) is suspended from m1, and a second mass m2 is suspended from the second spring's lower end. Assuming that the masses move only in a vertical direction and using coordinates y1 and y2 measured from the masses' equilibrium positions, show that the equations of motion can be written in the matrix form My=Ky, where y is the 2×1 column made up of y1 and y2. Find the 2×2 matrices M and K.

Two equal masses m are constrained to move without friction, one on the positive x axis and one on the positive y axis. They are attached to two identical springs (force constant k ) whose other ends are attached to the origin. In addition, the two masses are connected to each other by a third spring of force constant k. The springs are chosen so that the system is in equilibrium with all three springs relaxed (length equal to unstretched length). What are the normal frequencies? Find and describe the normal modes.

Consider a frictionless rigid horizontal hoop of radius R. Onto this hoop I thread three beads with masses 2m,m, and m, and, between the beads, three identical springs, each with force constant k. Solve for the three normal frequencies and find and describe the three normal modes.

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