Figure 7.12 shows a crude model of a yoyo. A massless string...

Figure 7.12 shows a crude model of a yoyo. A massless string is suspended vertically from a fixed point and the other end is wrapped several times around a uniform cylinder of mass $m$ and radius $R$. When the cylinder is released it moves vertically down, rotating as the string unwinds. Write down the Lagrangian, using the distance $x$ as your generalized coordinate. Find the Lagrange equation of motion and show that the cylinder accelerates downward with $\ddot{x}=2 g / 3$. [Hints: You need to remember from your introductory physics course that the total kinetic energy of a body like the yoyo is $T=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2},$ where $v$ is the velocity of the center of mass, $I$ is the moment of inertia (for a unform cylinder, $I=\frac{1}{2} m R^{2}$ ) and $\omega$ is the angular velocity about the CM. You can express $\omega$ in terms of $\dot{x}.$]

Key Concepts

Constraint Linking Translation and Rotation

Constraints that relate translational and rotational motion, such as a relationship between the linear displacement of a body's center of mass and its angular displacement, are essential in simplifying the analysis of systems with combined modes of motion. These constraints reduce the number of independent variables and are key to accurately formulating and solving the system's dynamics.

Euler-Lagrange Equation

The Euler-Lagrange equation is a set of differential equations derived from the Lagrangian of a system, establishing that the time derivative of the partial derivative of the Lagrangian with respect to the generalized velocity equals the partial derivative of the Lagrangian with respect to the generalized coordinate. This equation is fundamental for obtaining the equations of motion for systems described by generalized coordinates.

Potential Energy

Potential energy in the context of gravitational systems is associated with the position of a mass within a gravitational field, typically modeled as mgh or a similar function of vertical displacement. Including this term in the Lagrangian allows one to account for the forces acting on the system due to gravity.

Generalized Coordinates

Generalized coordinates are the set of parameters that uniquely describe the configuration of a system, taking into account its constraints. They are chosen to simplify the problem, often reducing the number of independent variables required to describe the system's motion, and are not limited to standard Cartesian coordinates.

Lagrangian Mechanics

Lagrangian mechanics is an approach in classical mechanics that formulates the dynamics of a system in terms of the difference between its kinetic and potential energies. This framework, by using the Lagrangian function, enables one to derive the equations of motion through the application of the Euler-Lagrange equations, making it especially useful for systems with constraints or complex interactions.

Kinetic Energy (Translational and Rotational)

In systems involving both translation and rotation, the total kinetic energy is the sum of translational kinetic energy, given by ½ m v², and rotational kinetic energy, expressed as ½ I ?², where I is the moment of inertia and ? is the angular velocity. This decomposition is crucial for correctly formulating the energy balance in the Lagrangian framework.

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