Figure 7.16 is a bird's-eye view of a smooth horizontal wire...

Figure 7.16 is a bird's-eye view of a smooth horizontal wire hoop that is forced to rotate at a fixed angular velocity $\omega$ about a vertical axis through the point $A$. A bead of mass $m$ is threaded on the hoop and is free to move around it, with its position specified by the angle $\phi$ that it makes at the center with the diameter $A B$. Find the Lagrangian for this system using $\phi$ as your generalized coordinate. (Read the hint in Problem $7.29 .$ ) Use the Lagrange equation of motion to show that the bead oscillates about the point $B$ exactly like a simple pendulum. What is the frequency of these oscillations if their amplitude is small?

Key Concepts

Small Oscillation Approximation

The small oscillation approximation is used to simplify the analysis of a system's behavior near a stable equilibrium point. By assuming that deviations from equilibrium are small, the equations of motion can be linearized, leading to a simple harmonic oscillator model. This approximation makes it easier to derive characteristics like the oscillation frequency, which is particularly useful in systems that mimic a pendulum or other simple oscillatory systems.

Rotating Reference Frames

Rotating reference frames are used to analyze dynamics from the perspective of a system that is itself rotating. In such frames, additional inertial effects such as Coriolis and centrifugal forces must be considered, as they can significantly influence the motion of objects, particularly in systems where elements move relative to the rotating frame.

Kinetic and Potential Energy Contributions

In constructing the Lagrangian for a system, the kinetic energy includes not only the standard translational motion but also contributions from rotational motion or the motion induced by the rotating support. The potential energy arises from conservative forces, and properly identifying these energy components is crucial for setting up the Lagrangian correctly, especially in systems with combined rotational and translational dynamics.

Lagrangian Mechanics

Lagrangian mechanics is an approach to classical mechanics that reformulates the equations of motion in terms of a scalar quantity called the Lagrangian, which is the difference between the kinetic and potential energies of the system. This method provides a powerful framework for dealing with complex systems and constraints by applying the principle of stationary action and using the Euler-Lagrange equations to derive the equations of motion.

Generalized Coordinates

Generalized coordinates are a set of parameters that uniquely describe the configuration of a system after accounting for its constraints. These coordinates simplify the analysis of mechanical systems by reducing the number of variables needed to describe the system's state, allowing the formulation of the Lagrangian in terms of these minimal independent variables.

Transcript

00:01 So we know that theta, the angle measure, will be equal to the amplitude times the exponential function of negative gamma t times cosine of omega prime t.

00:11 The exponential function simply means that whatever is inside the parentheses is a power with a base of e.

00:19 We know that a, the amplitude, is 15 degrees, and that t is 8 seconds.

00:27 So 8 seconds have elapsed.

00:28 For part a, we know that 5 .5 degrees will equal 15 degrees times the exponential function of negative 8 .0 seconds times gamma.

00:45 And so solving for gamma, gamma will simply be equal to negative 1 divided by 8 times the natural log of 5 .5 divided by 15.

00:58 And so gamma is going to be equal to positive 0 .1254 per second.

01:08 For part b, they want us to find the period.

01:13 So we can say that omega -0, the original angular velocity, would be equal to mgh over the moment of inertia.

01:24 And this is going to be equal to mg times 1 over 2l divided by the moment of inertia.

01:33 In this case, it would be one third ml squared.

01:37 And this is giving us the square root of 3g over 2l.

01:43 Now, we can find the formula for the damped frequency, the damped frequency, or we can say omega prime, would be equal to the square root of the original angular velocity squared minus gamma squared, all to this one -half power.

02:06 So this is going to give us the square root of 3g over 2 times l minus gamma squared...

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