Question

(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 $y_1$ and $y_2,$ 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

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

Step-by-step solution

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), $y_1$ and $y_2$ (sideways displacements).

Set Up Generalized Coordinates

Let $x$ be the longitudinal displacement of the entire rod. The sideways movements of the two ends are noted as $y_1$ and $y_2$. The rod turns slightly, introducing a small angle $\theta$.

Determine the Potential Energy

The potential energy for the system can be expressed in terms of the heights of the two ends: $V=\frac{1}{2}mg(2l-\text{(height differences of ends)})$, where $m$ is the mass of the rod and $g$ is the gravitational constant.

Calculate the Kinetic Energy

The kinetic energy is the sum of translational and rotational energies: $$T=\frac{1}{2}m{\dot{x}}^2+\frac{1}{2}m({(\frac{l}{2}\dot{\theta }+\dot{y_1})}^2+{(\frac{l}{2}\dot{\theta }+\dot{y_2})}^2).$$

Formulate the Lagrangian

The Lagrangian $L$ is calculated by subtracting the potential energy from the kinetic energy: $L=T-V$.

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.

Find the Normal Modes and Frequencies

Analyze the equations found by assuming solutions of the form: $x=A{e}^{i\omega t}$, $y_1=B{e}^{i\omega t}$, and $y_2=C{e}^{i\omega t}$. This leads to a matrix equation $M{\omega }^2=K$, from which you can solve for $\omega$ to find normal frequencies.

Key concepts

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: $x$, $y_1$, and $y_2$, which are strategically chosen to make calculations straightforward
• The $x$ coordinate represents longitudinal movement, while $y_1$ and $y_2$ 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.