Question

Consider a frictionless rigid horizontal hoop of radius \(R\). Onto this hoop I thread three beads with masses \(2 m, m,\) 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.

Short answer

The three normal frequencies are found by solving the characteristic equation for \( \omega \), and normal modes describe the oscillation patterns.

Step-by-step solution

Understanding the System Layout

Consider the physical system as a circular loop of radius \( R \) with three beads positioned at angles \( \theta_1, \theta_2, \theta_3 \), each separated by identical springs with spring constant \( k \). The masses of the beads are \( 2m, m, \) and \( m \). The springs exert restoring forces when the beads are displaced from their equilibrium positions.

Setting Up Coordinates

Define positions of the beads along the hoop by angles \( \theta_1, \theta_2, \theta_3 \). Small oscillations lead to small displacements \( \Delta\theta_i \) from the equilibrium positions. Choosing angular coordinates simplifies the potential energy and kinetic energy expressions.

Formulate the Equations of Motion

Using small-angle approximations, the potential energy stored in each spring is expressed as \( U = \frac{1}{2} k R^2 (\Delta \theta_i - \Delta \theta_{i+1})^2 \). Write down the kinetic energy as \( T = \frac{1}{2} m_i R^2 \dot{\theta_i}^2 \) for each mass. Apply Lagrange's equations to derive the motion equations.

Construct the Matrix Equation

From the equations of motion, identify the terms that describe mass interactions. Assemble them into a matrix form such as \( M \ddot{x} + K x = 0 \), where \( M \) is the mass matrix and \( K \) is the stiffness matrix derived from the potential energy terms.

Solving for Normal Frequencies

Solve the characteristic equation \( \det(K - \omega^2 M) = 0 \) for \( \omega \), where \( \omega \) represents angular frequencies. This results in a three-degrees-of-freedom system characterized by a cubic equation in \( \omega^2 \). Solving this yields the normal frequencies.

Determining Normal Modes

For each normal frequency \( \omega_i \), substitute back into the matrix equation to solve for the displacement vectors that describe the normal modes. The vectors represent patterns of motion where all parts of the system oscillate with the same frequency.

Key concepts

Normal Modes

Normal modes are specific patterns of motion that occur in a system where all components oscillate at the same frequency. They are crucial in understanding how complex systems move. In this system involving a circular hoop and beads, normal modes represent the motion patterns of the three beads connected by springs.

• The beads oscillate in such a way that their motion maintains a synchronized frequency for each mode.
• These modes exist due to the symmetry and constraints of the system, allowing specific frequencies to emerge naturally.
• The study of normal modes helps predict the behavior of the system when it is disturbed from its equilibrium.

By considering only these modes, the problem can be simplified and solutions can be found using mathematical methods, such as solving characteristic equations. This makes it easier to understand complex oscillations in mechanical systems.

Lagrange's Equations

Lagrange's equations offer a powerful approach to understanding the motion of systems, especially when dealing with multiple degrees of freedom like in our bead and hoop system. These equations provide a way to derive the equations of motion by considering the system's kinetic and potential energies, rather than directly using forces.

• The Lagrangian function, defined as the difference between kinetic and potential energy, serves as a foundational element in this approach, and it is formulated as \( L = T - U \).
• The Euler-Lagrange equation, \( \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 \), where \( q_i \) are the generalized coordinates, is used to derive the system's equations of motion.
• This method is particularly useful for systems with constraints, such as our hoop, which are naturally incorporated into the formulation.

Using Lagrange's equations, it's easier to handle the small oscillations of the beads and arrive at a set of equations that describe their collective motion.

Small Oscillations

Small oscillations refer to slight deviations from an equilibrium position where linear approximations can be accurately used to describe motion. These are common in systems where components are bound to specific paths, such as the beads on a hoop.

• When analyzing this system, small oscillations allow the use of the small angle approximation, simplifying trigonometric functions like sine and cosine, which helps to linearize the mathematical models.
• The displacement from the equilibrium position is small enough that forces can be approximated linearly, described as \( F \approx -kx \), where \( k \) is the spring constant and \( x \) the displacement.
• This simplification leads to more manageable equations, facilitating solutions by making the system's matrix equations easier to solve.

Understanding small oscillations helps in forming the basic assumptions that guide the analysis and solutions of normal mode problems.

Eigenfrequencies

Eigenfrequencies are the characteristic frequencies at which a system naturally oscillates. In the context of normal modes, each eigenfrequency corresponds to a specific normal mode of the system.

• In our beads and hoop system, finding these frequencies involves solving the characteristic equation derived from the matrix formulation, \( \det(K - \omega^2 M) = 0 \).
• These frequencies are determined by the physical properties of the system, such as mass distribution and stiffness (spring constants).
• The solutions to the characteristic equation are the different values of \( \omega \), which provide insights into possible patterns of oscillation.

Understanding eigenfrequencies is crucial because they indicate how the system will respond to external disturbances, showing stability or potential resonances under specific conditions.