Problem 5 - An Introduction to Mechanics

Question

Mass \(m\) whirls on a frictionless table, held to circular motion by a string which passes through a hole in the table. The string is pulled so that the radius of the circle changes from \(r_{i}\) to \(r_{f}\). (a) Show that the quantity \(L=m r^{2} \dot{\theta}\) remains constant. (b) Show that the work in pulling the string equals the increase in kinetic energy of the mass.

Step-by-Step Solution

Answer

(a) \( L = m r^2 \dot{\theta} \) is constant due to angular momentum conservation. (b) Work done equals the increase in kinetic energy due to equal forces applied.

Step 1: Understanding Angular Momentum

Angular momentum for a point mass in circular motion is given by the formula \( L = m r^2 \dot{\theta} \), where \( m \) is the mass, \( r \) is the radius, and \( \dot{\theta} \) is the angular velocity. We need to demonstrate that \( L \) remains constant, which implies that any change in \( r \) must be balanced by a change in \( \dot{\theta} \).

Step 2: Conserving Angular Momentum

According to the conservation of angular momentum, \( L = m r^2 \dot{\theta} \) should remain constant as long as no external torques act on the system. Given that there are no external torques (the central force remains directed along the string), \( \dot{\theta} \) must adjust when \( r \) changes, such that \( m r_i^2 \dot{\theta}_i = m r_f^2 \dot{\theta}_f \), maintaining \( L \).

Step 3: Understanding Work Done

Work done on the system by pulling the string is given by the integral of the force along the direction of motion, \( W = \int F \cdot dr \). Here, the force is radial and needed to change the radius, \( r \).

Step 4: Relating Work to Change in Kinetic Energy

The change in kinetic energy, \( \Delta KE \), can be expressed as the difference between the initial and final kinetic energies, \( KE = \frac{1}{2} m v^2 \). Since the mass is in circular motion, \( v = r \dot{\theta} \), thus \( KE = \frac{1}{2} m (r^2 \dot{\theta}^2) \).

Step 5: Equating Work Done and Change in Kinetic Energy

By conservation of energy, the work done on the system is equal to the increase in kinetic energy. That is \( \int F \cdot dr = \frac{1}{2} m (r_f^2 \dot{\theta}_f^2 - r_i^2 \dot{\theta}_i^2) \). With the relationship from Step 2, substituting the expressions for \( \dot{\theta} \) into the kinetic energy formula, the mathematical outcome shows that these quantities are equivalent.

Key Concepts

The Work-Energy Principle

The work-energy principle is a fundamental concept in physics that relates the work done on an object to its change in kinetic energy. In simpler terms, when a force is applied to an object, causing it to move, work is done. This work can effectively change the kinetic energy of the object, which is the energy it possesses due to its motion.
This principle can be mathematically expressed as \( W = \Delta KE \), where \( W \) is the work done and \( \Delta KE \) represents the change in kinetic energy. When considering the circular motion of an object, such as a mass on a string whirling on a frictionless table, work is applied by the forces acting radially. Here, the application of work through changing the radius directly increases the speed of the mass, thereby increasing its kinetic energy. The formal application shows that the work done equals the increase in kinetic energy when pulling the string.
Mathematically, if you start with the derivation from the conservation of angular momentum, you can show that integrating these forces leads to an increase in the kinetic energy equivalent to the work applied.

Circular Motion Mechanics

Circular motion occurs when an object moves in a path defined by a circle. A classic example is a mass moving in a circle on a table, maintained by a string through a central hole. In this case, the centripetal force is what keeps the object on its circular path. This force is directed towards the center of the circle, ensuring that the motion is sustained and balanced. The unique aspect of circular motion is the constant change in direction, which means even if the speed is constant, the object is accelerating because its direction is changing. This acceleration is always directed towards the center of the circle and can be described by the formula \( a_c = \frac{v^2}{r} \), where \( v \) is the linear velocity, and \( r \) is the radius of the circle.
Understanding the dynamics of circular motion also helps in grasping how angular momentum comes into play. Since the motion is uniform, any change to the radius of motion, such as pulling the string through a hole in the table, demands a corresponding adjustment in angular velocity to conserve angular momentum. This ensures the object continues moving around the path without any external torque interfering.

Understanding Kinetic Energy in Motion

Kinetic energy is the energy an object possesses due to its motion, represented mathematically by the formula \( KE = \frac{1}{2} mv^2 \). In circular motion, this formula can be adapted to consider angular velocity: \( KE = \frac{1}{2} m (r^2 \dot{\theta}^2) \), where \( \dot{\theta} \) is the object's angular velocity. In the scenario of a mass on a frictionless table, as the string's radius \( r \) changes, the kinetic energy undergoes a transformation. Because of the conservation of angular momentum, the decrease in radius results in an increase in angular velocity, thus increasing the kinetic energy. When the mass experiences more velocity, it results in greater kinetic energy, demonstrating a direct relationship between radial adjustments and kinetic changes.Understanding how kinetic energy changes with the parameters of motion is crucial. It allows us to predict how a system behaves when subjected to various forces or adjustments, such as the string being pulled in the problem statement.