Question

A particle of mass \(m\) is moving on a frictionless horizontal table and is attached to a massless string, whose other end passes through a hole in the table, where I am holding it. Initially the particle is moving in a circle of radius \(r_{\mathrm{o}}\) with angular velocity \(\omega_{\mathrm{o}},\) but I now pull the string down through the hole until a length \(r\) remains between the hole and the particle. What is the particle's angular velocity now?

Short answer

The new angular velocity \( \omega \) is \( \frac{r_o^2 \omega_o}{r^2} \).

Step-by-step solution

Identify Initial Angular Momentum

The initial angular momentum of the particle can be expressed as the product of its moment of inertia and angular velocity. Initially, the particle is moving in a circle of radius \( r_o \) with angular velocity \( \omega_o \). The moment of inertia \( I_i \) at this point is \( m r_o^2 \). Therefore, the initial angular momentum \( L_i \) is given by: \[ L_i = I_i \omega_o = m r_o^2 \omega_o. \]

Express Final Angular Momentum

After pulling the string down, the remaining length of the string is \( r \), and the new moment of inertia \( I_f \) is \( m r^2 \). If the new angular velocity is \( \omega \), the final angular momentum \( L_f \) is: \[ L_f = I_f \omega = m r^2 \omega. \]

Apply Conservation of Angular Momentum

Since no external torques are acting on the system, the angular momentum is conserved. This gives: \[ L_i = L_f. \] Substituting the expressions from Steps 1 and 2: \[ m r_o^2 \omega_o = m r^2 \omega. \]

Solve for Final Angular Velocity

To find the particle's angular velocity \( \omega \) after pulling the string, we solve the equation: \[ m r_o^2 \omega_o = m r^2 \omega \rightarrow \omega = \frac{r_o^2 \omega_o}{r^2}. \]

Key concepts

Moment of Inertia

Moment of inertia is a fundamental concept when it comes to understanding rotational motion. It is the rotational equivalent of mass in linear motion. In simple terms, it measures how much an object resists changes in its rotational state. Think of it as the rotational "mass," determining how easily an object can spin.

The formula for the moment of inertia depends on the mass distribution of an object. For a particle moving in a circle, like in this exercise, it can be calculated as the product of the mass and the square of the radius of motion:

• Moment of Inertia, \( I = m imes r^2 \) where \( m \) is the mass and \( r \) is the radius.

When you decrease the radius by pulling the string, the moment of inertia decreases since it depends on \( r^2 \). This change affects how the particle rotates, as seen in the principle of angular momentum conservation.

Angular Velocity

Angular velocity is an essential part of understanding how quickly an object is rotating. It tells us how fast something spins around a central point, measured in radians per second.

In the given exercise, the initial angular velocity \( \omega_o \) describes the rate at which the particle moves in its original circular path. With the string's length pulled to a smaller radius, the angular velocity changes to maintain the balance of angular momentum. As the radius decreases, the angular velocity must increase to conserve the system's rotational motion due to the reduced moment of inertia.

Here's how you can determine the new angular velocity:

• Initial angular velocity: \( \omega_o \)
• New angular velocity: \( \omega = \frac{r_o^2 \omega_o}{r^2} \)

This equation shows the inverse relationship between radius and angular velocity, driven by the conservation of angular momentum.

Circular Motion

Circular motion occurs when an object moves continuously along a circular path. It's characterized by the motion's radius, speed (or velocity), and the force keeping it on its circular path.

In this exercise, the particle is initially moving in a uniform circular path. The key forces at play here are centripetal, meaning they act towards the center of the circle, maintaining the particle's path.

When the string is pulled, not only does the path radius change, but so do other parameters:

• The centripetal force adjusts, but its nature stays the same - always towards the circle's center.
• The path becomes tighter or broader according to how the string length varies.
• The particle's speed and angular velocity adapt to maintain its motion without any external torques acting.

Understanding circular motion helps predict how objects will behave when their paths or conditions change, like in the exercise where pulling the string modifies the circle's size and hence the motion dynamics.