Question

A yo-yo (as shown) falls under gravity. Assume that it falls straight down, unwinding as it goes. Find the Lagrange equation of motion. Hints: The kinetic energy is the sum of the translational energy \(\frac{1}{2} m \dot{z}^{2}\) and the rotational energy \(\frac{1}{2} I \dot{\theta}^{2}\) where \(I\) is the moment of inertia. What is the relation between \(\dot{z}\) and \(\dot{\theta}\) ? Assume the yo-yo is a solid cylinder with inner radius \(a\) and outer radius \(b\).

Short answer

The Lagrange equation of motion is \[ m (a^2 + \frac{1}{2} b^2) \ddot{\theta} + mga = 0 \].

Step-by-step solution

Define the variables and given quantities

Let the mass of the yo-yo be denoted by \(m\), the distance fallen by \(z\), the angular displacement by \(\theta\), the outer radius by \(b\), and the inner radius by \(a\).

Establish the relationship between translational and rotational motion

Since the yo-yo unwinds as it falls, the linear speed \(\dot{z}\) and the angular speed \(\dot{\theta}\) are related by the inner radius \(a\) as follows: \[ \dot{z} = a \dot{\theta} \]

Write the kinetic energy expressions

The total kinetic energy (T) is the sum of the translational kinetic energy and the rotational kinetic energy. The translational kinetic energy is \( \frac{1}{2} m \dot{z}^2 \). The rotational kinetic energy is \( \frac{1}{2} I \dot{\theta}^2 \), where \(I\) is the moment of inertia of the solid cylinder, given by \[ I = \frac{1}{2} m b^2 \].

Substitute the moment of inertia and the relationship between \(\dot{z}\) and \(\dot{\theta}\)

Substitute the moment of inertia \( I = \frac{1}{2} m b^2 \) into the rotational kinetic energy expression: \[ \frac{1}{2} I \dot{\theta}^2 = \frac{1}{2} \left( \frac{1}{2} m b^2 \right) \dot{\theta}^2 = \frac{1}{4} m b^2 \dot{\theta}^2 \]. Also substitute \( \dot{z} = a \dot{\theta} \) into the translational kinetic energy expression: \[ \frac{1}{2} m \dot{z}^2 = \frac{1}{2} m (a \dot{\theta})^2 = \frac{1}{2} m a^2 \dot{\theta}^2 \].

Write the total kinetic energy

Combine the translational and rotational kinetic energy expressions to get the total kinetic energy \( T \): \[ T = \frac{1}{2} m a^2 \dot{\theta}^2 + \frac{1}{4} m b^2 \dot{\theta}^2 = \frac{1}{2} m (a^2 + \frac{1}{2} b^2) \dot{\theta}^2 \].

Write the potential energy

The potential energy (V) of the yo-yo when it has fallen a distance \(z\) under gravity \(g\) is given by: \[ V = mgz \].

Formulate the Lagrangian

The Lagrangian \(L\) is the difference between the kinetic energy and the potential energy: \[ L = T - V = \frac{1}{2} m (a^2 + \frac{1}{2} b^2) \dot{\theta}^2 - mgz \].

Express \(z\) in terms of \(\theta\)

Since \( \dot{z} = a \dot{\theta} \), we can write \(z\) as \(z = a \theta\). Substitute this into the Lagrangian: \[ L = \frac{1}{2} m (a^2 + \frac{1}{2} b^2) \dot{\theta}^2 - mga \theta \].

Compute the equations of motion using the Euler-Lagrange equation

The Euler-Lagrange equation is given by: \[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) - \frac{\partial L}{\partial \theta} = 0 \]. Compute the partial derivatives: \( \frac{\partial L}{\partial \dot{\theta}} = m (a^2 + \frac{1}{2} b^2) \dot{\theta} \) and \( \frac{\partial L}{\partial \theta} = -mga \). Substitute into the Euler-Lagrange equation: \[ \frac{d}{dt} (m (a^2 + \frac{1}{2} b^2) \dot{\theta}) + mga = 0 \]. Simplify to get the equation of motion: \[ m (a^2 + \frac{1}{2} b^2) \ddot{\theta} + mga = 0 \].

Key concepts

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. For the yo-yo, there are two types of kinetic energy: translational and rotational.
The translational kinetic energy pertains to the linear movement when the yo-yo falls straight down. Its formula is given by \(\frac{1}{2} m \dot{z}^{2}\), where \(m\) is the mass and \(\dot{z}\) is the linear velocity.
Rotational kinetic energy, on the other hand, is associated with the spinning motion of the yo-yo. This is given by \(\frac{1}{2} I \dot{\theta}^{2}\). Here, \(\mathrm{I}\) is the moment of inertia, and \(\dot{\theta}\) is the angular velocity.
Combining both, you get the total kinetic energy of the yo-yo.

Potential Energy

Potential energy represents the energy stored due to the position of an object in a gravitational field. For a yo-yo falling under gravity, the potential energy depends on how far it has fallen.
The potential energy is given by \(\frac{1}{2} V = mgz\), where \(m\) is the mass of the yo-yo, \(g\) is the acceleration due to gravity, and \(z\) is the distance fallen.
As the yo-yo descends, it loses potential energy, which converts into kinetic energy (both translational and rotational) due to the conservation of energy principle.

Euler-Lagrange Equation

The Euler-Lagrange equation is a fundamental equation in physics that is derived from the principle of least action. It helps to find the equations of motion for a system.
The equation is given by \[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) - \frac{\partial L}{\partial \theta} = 0 \]. Here, \(L\) denotes the Lagrangian, which is the difference between kinetic energy (K) and potential energy (V).
By applying this equation, we can derive the differential equations that describe the system's motion. For the yo-yo, the Lagrangian is formulated in terms of its kinetic and potential energies, leading to its equation of motion.

Translational Motion

Translational motion refers to the linear movement of an object from one point to another. For the yo-yo, this is the motion as it falls straight downward.
The key variable for translational motion is the distance fallen, denoted by \(z\). The velocity of this motion is represented by \(\dot{z}\), the time derivative of \(z\).
Understanding the translational motion helps in separating the overall kinetic energy into its linear and rotational components, which are then combined to get the total kinetic energy.

Rotational Motion

Rotational motion refers to the spinning of the yo-yo around its axis. This type of movement introduces rotational kinetic energy.
The essential variables are the angular displacement \(\theta\) and angular velocity \(\dot{\theta}\).
For a solid cylinder yo-yo, the moment of inertia \(I\) is given by \(\frac{1}{2} m b^{2}\), where \(b\) is the radius of the yo-yo. This helps in finding the rotational kinetic energy with the formula \(\frac{1}{2} I \dot{\theta}^{2}\).
Relating \(\dot{z}\) to \(\dot{\theta}\) using the radius \(a\), where \(\dot{z}=a\dot{\theta}\), is key to combining the different forms of kinetic energy in the Lagrangian formulation.