Question

A mass \(m\) lies on a table and is connected to a string of length \(\ell\) as shown in Figure 10.45. The string passes through a hole in the table and is connected to another mass \(M\) that is hanging in the air. We assume that the string remains taught and that mass \(M\) can only move vertically. a. The Lagrangian for this setup is $$ \mathcal{L}=\frac{1}{2} M \dot{r}^{2}+\frac{1}{2} m\left(\dot{r}^{2}+r^{2} \dot{\theta}^{2}\right)+M g(\ell-r) $$ where \(r\) and \(\theta\) are polar coordinates describing where mass \(m\) is on the table with respect to the hole. Explain why the terms in the Lagrangian are appropriate. b. Derive the equations of motion for \(r(t)\) and \(\theta(t)\). c. What angular velocity is needed for mass \(m\) to maintain uniform circular motion?

Short answer

The terms in the Lagrangian represent the kinetic energy of the two masses and the potential energy of the hanging mass 'M'. The Euler-Lagrange equation is used to derive the equations of motion for \( r(t) \) and \( \theta(t) \). Uniform circular motion is achieved when the net radial force is zero, by substituting this condition into the \( r \) equation of motion, the required angular velocity for maintaining uniform circular motion is obtained.

Step-by-step solution

Understanding the Lagrangian

The Lagrangian function, represented as \( \mathcal{L} \), is defined as the kinetic energy subtracted by the potential energy of a system. In our specific case, the kinetic energy of the system is the kinetic energy of the two masses. The potential energy is the gravitational potential energy of the hanging mass 'M'. The potential energy changes as 'M' moves up or down, which is represented by \( \ell - r \). Thus each term in the Lagrangian represents an aspect of the energy of the system.

Deriving the equations of motion

The equation of motion can be obtained using the Euler-Lagrange equation, which states that derivative of Lagrangian with respect to the general coordinates minus its derivative with respect to the time derivative of the general coordinates must be equal to zero. Two equations can be obtained by applying this to the two general coordinates \( r \) and \( \theta \).

Deriving expression for angular velocity

For uniform circular motion, the net radial force is zero. Applying this condition to the \( r \) equation of motion will give the expression for angular velocity.

Key concepts

Equations of Motion

Deriving equations of motion is a crucial part in analyzing any system using Lagrangian mechanics. In this scenario, we have two masses and a string passing through a hole. The Lagrangian, \[ \mathcal{L} = \frac{1}{2} M \dot{r}^{2} + \frac{1}{2} m (\dot{r}^{2} + r^{2} \dot{\theta}^{2}) + M g(\ell - r) \]is used to derive the equations describing the motion of the masses. The process involves the Euler-Lagrange equation:\[ \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}} \right) - \frac{\partial \mathcal{L}}{\partial q} = 0 \]where \( q \) represents the generalized coordinates, namely \( r \) and \( \theta \) in this problem. Calculating these for each coordinate:

• For \( r(t) \), we consider terms involving radial motion and potential energy.
• For \( \theta(t) \), we focus on angular motion about the hole.

Both lead to distinct equations capturing how each coordinate evolves over time.

Kinetic Energy

Kinetic energy in a system is related to the motion of its components. In this exercise, the system consists of two masses: one on the table and one hanging. The kinetic energy forms part of the total energy described by the Lagrangian. To break it down:

• The kinetic energy of mass \( M \) hanging in the air contributes \( \frac{1}{2} M \dot{r}^2 \), reflecting its vertical motion.
• Mass \( m \) on the table has two components of motion: radial and angular. Its kinetic energy is \( \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) \), which accounts for both such movements.

Kinetic energy is crucial as it captures how the position and velocity of the masses change over time. It also plays an integral role in forming the Lagrangian, allowing for the derivation of accurate equations of motion.

Potential Energy

Potential energy is energy stored by virtue of the position or configuration of objects in a system. In this context, potential energy originates from the gravitational interaction involving mass \( M \). As \( M \) moves vertically, the potential energy changes with its height.The Lagrangian includes a term for this potential energy: \[ M g (\ell - r) \]where \( \ell - r \) represents the distance from the tabletop to the hanging mass. Key points to consider:

• This term decreases as \( r \) increases because the hanging mass is moving upwards.
• This energy is crucial for understanding the effects of gravity on the system and how it impacts the dynamics, such as tension in the string.

Changes in potential energy often result in changes in system dynamics, influencing motion significantly.

Polar Coordinates

Polar coordinates offer a convenient method for analyzing systems with rotational or radial symmetry, such as the one in this exercise. Polar coordinates are denoted by \( r \) (radius) and \( \theta \) (angle), and they prove beneficial in simplifying complex problems involving circular paths.In this scenario:

• \( r \) provides the radial distance from the hole in the table to the position of mass \( m \).
• \( \theta \) represents the angle showing the orientation of \( m \) relative to the reference direction, which in this case is likely aligned with some fixed axis.

Utilizing polar coordinates allows for a more straightforward application of the Lagrangian approach, as it inherently captures aspects of circular motion. This makes it easier to derive equations of motion for systems like the one here, where an object moves in a plane about a central point.