Question

${}^{†}$ A uniform rod $AB$ of mass $m$ and length $2a$ is suspended by two light inextensible strings $PA$ and $QB$, each of length $2a$. The fixed points $P$ and $Q$ are at the same level, distance $2a$ apart; and in equilibrium, $PA$ and $QB$ are vertical. The system performs small oscillations about its equilibrium configuration. Determine the normal frequencies and describe the normal modes.

Short answer

Question: Determine the normal frequencies and normal modes of a given system consisting of a uniform rod and two inextensible strings. Solution: 1. Calculate the potential energy (U) and kinetic energy (T) of the system using the angles α and β formed by the strings at points A and B respectively with the vertical. 2. Compute the Lagrangian (L) of the system as the difference between kinetic (T) and potential (U) energy. 3. Utilize the Euler-Lagrange equations to obtain the equations of motion for α and β. 4. Assume α and β to be harmonic functions in the form of e^(iωt) and substitute into the equations of motion. Solve the characteristic equations for ω to obtain the normal frequencies. 5. Find the normal modes by substituting the obtained normal frequencies back into the assumed harmonic functions for α and β and express them in terms of the equilibrium position of the rod and the angles α and β.

Step-by-step solution

Determine Potential and Kinetic Energy of the System

Firstly, let's denote the angles at which the strings $PA$ and $QB$ make with the vertical as $\alpha$ and $\beta$, respectively. The potential energy of the system is due to the gravitational force acting on the rod that is at the center of mass. The height of the center of the rod is: $$h=a\cos (\alpha )+a\cos (\beta ).$$ Since the gravitational potential energy is given by $U=mgh$, the potential energy of the system is: $$U=mga(\cos \alpha +\cos \beta ).$$ Now, let's determine the kinetic energy of the system. The kinetic energy of the rod can be represented as the sum of the kinetic energy of each end ($A$ and $B$) of the rod: $$T=\frac{1}{2}m({\dot{x}}_A^2+{\dot{y}}_A^2)+\frac{1}{2}m({\dot{x}}_B^2+{\dot{y}}_B^2),$$ where ${\dot{x}}_A,{\dot{y}}_A,{\dot{x}}_B,\text{and}{\dot{y}}_B$ are the derivatives of the positions of $A$ and $B$.

Compute the Lagrangian

Next, we need to find the Lagrangian of the system ($L$) which is given by the difference between the kinetic energy ($T$) and the potential energy ($U$). $$L=T-U.$$

Calculate the Equations of Motion

We will use the Euler-Lagrange equations to derive the equations of motion for $\alpha$ and $\beta$. The Euler-Lagrange equations are given by: $$\frac{d}{dt}\frac{\partial L}{\partial \dot{\alpha }}-\frac{\partial L}{\partial \alpha }=0,$$ $$\frac{d}{dt}\frac{\partial L}{\partial \dot{\beta }}-\frac{\partial L}{\partial \beta }=0.$$ Solving these equations will give us the equations of motion for the given system in terms of $\alpha$ and $\beta$.

Find the Normal Frequencies and Normal Modes

To find the normal frequencies of the system we will assume $\alpha$ and $\beta$ are harmonic functions of the form $e^{i\omega t}$ where $\omega$ is the normal frequency. After substituting these assumed functions into the equations of motion, we will obtain characteristic equations for $\omega$. Solving these equations will give us the normal frequencies of the system. Finally, to find the normal modes, we will substitute the obtained normal frequencies back into the assumed harmonic functions for $\alpha$ and $\beta$ and express the normal modes in terms of the equilibrium position of the rod and the angles $\alpha$ and $\beta$.

Key concepts

Lagrangian Mechanics

Lagrangian mechanics, a reformulation of classical mechanics, is particularly powerful when dealing with complex systems like our oscillating uniform rod. It provides a streamlined approach to finding equations of motion, especially in systems with constraints. At the heart of this framework is the Lagrangian function, represented as $L=T-U$, where $T$ is the kinetic energy and $U$ is the potential energy of the system.

In our exercise, we define $T$ by considering the kinetic energy of each end of the rod and $U$ as the gravitational potential energy of the rod based on the height of its center of mass. The beauty of Lagrangian mechanics is in how it elegantly uses the mathematically convenient Lagrangian function to encapsulate all the dynamical information of a system. When articulated through the Euler-Lagrange equations, it leads directly to the equations of motion without invoking force vectors explicitly, simplifying the process of analyzing the system's behavior.

Small Oscillations

When we look at small oscillations, we're focusing on systems that displace slightly from an equilibrium position and exhibit motion that can be approximated as linear — a critical simplification that makes the analysis more manageable. For our uniform rod suspended by strings, the small-angle approximation lets us view the motion as simple harmonic motion.

The key is to remember that small oscillations are a type of perturbation about a stable equilibrium. Hence, we consider the angles $\alpha$ and $\beta$ to be small enough to use linear approximations in our calculations. This approach helps in simplifying the trigonometric expressions in the potential energy and leads to a straightforward depiction of the system's dynamics. Small oscillations provide an essential gateway to understanding complex motion by breaking it down into simpler, linear components that are much easier to solve and interpret.

Normal Modes

Normal modes are patterns of motion in which all parts of the system oscillate sinusoidally with the same frequency, known as the normal frequency. These modes occur naturally in oscillating systems and are key for understanding the complete solution to the dynamics of our rod and strings system.

In the context of our exercise, once we derive the equations of motion using the Lagrangian method, the subsequent step is to assume solutions where $\alpha$ and $\beta$ change harmonically over time. The normal frequencies associated with these changes give us critical insights into how the system vibrates when slightly perturbed. By solving for these frequencies, we can describe the normal modes, which are essentially the 'pure' types of oscillations the system can exhibit without external forces acting to alter its motion. Understanding normal modes helps in predicting and controlling natural vibrations in various physical and engineering applications.