Problem 9 - Classical Dynamics of Particles and Systems

Question

A disk of mass \(M\) and radius \(R\) rolls without slipping down a plane inclined from the horizontal by an angle \(\alpha\). The disk has a short weightless axle of negligible radius. From this axis is suspended a simple pendulum of length \(l

Step-by-Step Solution

Answer

To find the equations of motion for the system, first define the generalized coordinates as \(x\) for the disk and \(\theta\) for the pendulum. Find the position of the pendulum bob in terms of these coordinates, and then determine the kinetic and potential energies of the system. The Lagrangian is given by \(L = T - V\). Apply Lagrange's equations of motion to the Lagrangian, resulting in two equations: \[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0\] \[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta}}\right) - \frac{\partial L}{\partial \theta} = 0\] Solve these equations to obtain the equations of motion for the system in terms of the generalized coordinates \(x\) and \(\theta\).

Step 1: Define the generalized coordinates

We need to define the generalized coordinates for both the disk (rolling down the incline) and the pendulum bob. Let's use the horizontal distance along the incline as the generalized coordinate for the disk, denoted by \(x\). Let the angle between the pendulum rod and the vertical be denoted by \(\theta\). Hence, the coordinates are \(\{x,\theta\}\).

Step 2: Finding the position of the pendulum bob

Let the position of the pendulum bob be represented by coordinates \((x_p, y_p)\). The position of the pendulum bob can be written as a function of \(x\) and \(\theta\). As the pendulum is suspended from the axle of the disk of radius R, we have: \[x_p = x + R\sin\theta - l \sin(\alpha - \theta)\] \[y_p = R(1 - \cos\theta) + l \cos(\alpha - \theta)\] Now we have the expressions for the position of the pendulum bob.

Step 3: Finding the kinetic and potential energies

To find the kinetic energy of the system, we need the velocities of both the disk and the pendulum bob. The kinetic energy, \(T\), contains contributions from the disk and the bob: \[T = \frac{1}{2}M(\dot{x}^2 + R^2\dot{\theta}^2) + \frac{1}{2}m(\dot{x}_p^2 + \dot{y}_p^2)\] The potential energy, \(V\), can be described as: \[V = Mgy_p + mgy_p\] Now we can find the Lagrangian of the system as \(L = T - V\).

Step 4: Applying Lagrange's equations of motion

To get the equations of motion, we will apply Lagrange's equations, which are given as: \[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q_i}}\right) - \frac{\partial L}{\partial q_i} = 0\] Here \(q_i\) represents the generalized coordinates \(\{x, \theta\}\). For the case of \(i=1\) (coordinate \(x\)), we find: \[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0\] For the case of \(i=2\) (coordinate \(\theta\)), we get: \[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta}}\right) - \frac{\partial L}{\partial \theta} = 0\] We need to solve these two equations to get the equations of motion for the system. These equations will describe the movement of the disk and the pendulum bob in terms of the generalized coordinates \(x\) and \(\theta\).

Key Concepts

Generalized Coordinates

Generalized coordinates are variables that help describe the configuration of a system in terms of its degrees of freedom. In the case of the disk and pendulum system, generalized coordinates are incredibly helpful because they reduce the complexity of the problem. Here, we choose two coordinates: the distance along the incline, denoted as \(x\), and the angle of the pendulum \(\theta\) relative to the vertical.

These coordinates allow us to track the states of movement for both the disk and the pendulum.

• \(x\) captures how far the disk has rolled along the inclined plane.
• \(\theta\) measures how much the pendulum swings from its resting vertical position.

By defining these coordinates, we simplify the problem significantly, focusing only on the essential movements without being bogged down by unnecessary details.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. For our system, both the disk and pendulum have kinetic energy contributions.

The disk's kinetic energy comes from its translational motion down the incline and its rotational motion as it rolls. This is represented by the term \(\frac{1}{2}M(\dot{x}^2 + R^2\dot{\theta}^2)\), capturing both its linear and angular velocities.

The pendulum's kinetic energy depends on the velocities of its bob in both horizontal \((\dot{x}_p^2)\) and vertical \((\dot{y}_p^2)\) directions. These velocities are affected by the disk's movement and the swinging of the pendulum itself. Hence, it is given by \(\frac{1}{2}m(\dot{x}_p^2 + \dot{y}_p^2)\).

• This term accounts for how quickly the pendulum swings and how it moves along with the disk.

Combining these, the total kinetic energy \(T\) reflects the whole system's dynamic motion.

Potential Energy

Potential energy is the energy that an object possesses due to its position relative to other objects. In this context, both the disk and the pendulum have gravitational potential energy as they are elevated above the ground.

For the system, the potential energy is represented as \(V = Mg y_p + mgy_p\). Since the disk and the bob of the pendulum both experience gravity, their heights above a reference point determine their potential energy.

• The term \(Mgy_p\) accounts for the gravitational pull on the disk as it rolls down the incline.
• The term \(mgy_p\) refers to the potential energy of the pendulum's bob as it swings through different vertical positions.

This energy changes as the system moves, influencing both the disk's path and the pendulum's swing.

Equations of Motion

The equations of motion are derived from Lagrange's equations and describe how a mechanical system evolves over time. These equations are central to understanding how the disk and pendulum will behave in motion.

To find these equations, we start with the Lagrangian, \(L = T - V\), where \(T\) is the total kinetic energy and \(V\) is the total potential energy.

Applying Lagrange's equations of motion, \[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0\] allows us to solve for each generalized coordinate \(q_i\), in this case, \(x\) and \(\theta\):

• For the disk's horizontal movement \(x\), the equation describes how it rolls down the plane.
• For the pendulum's angle \(\theta\), the equation accounts for its swing influenced by both gravity and the disk's motion.

These equations provide a complete picture of the system's dynamics and are necessary to predict future positions and velocities of both the disk and pendulum.